Advanced methods of glaciological modelling and time series analysis

A doctoral thesis by Aslak Grinsted

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Supervised by Docent John Moore Professor Sven-Erik Hjelt Opponent Dr. Martin Miles

Examiners Professor Ralf Greve Associate Professor Roderik van der Wal

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Abstract

This thesis covers a wide range of methods and research subjects and is thus broad in scope. The methods are applied to ice cores, Antarctic blue ice areas, sea level, and large scale climate variability.

Two new ice core proxies for continentality and summer melt were developed for the Lomonosovfonna ice core, central Svalbard. The melt proxy was based on the preferential washout different ions and the continentality proxy on the amplitudes of the annual signal in δ18O. These proxies suggest that summers in the Barents region were as warm (or warmer) than the present during the medieval warm period. A high quality chemical record of environmental changes was proven to be preserved in the core.

A simple and flexible flow model based on the continuity equation was developed with the purpose of dating the ancient surface ice in Antarctic blue ice areas. The model has been applied to three very different ice fields in East and West Antarctica.

The wavelet coherence and singular spectrum analysis methods were advanced and a phase-aware teleconnections method was developed. The methods were utilized in isolating a ~14 year quasi-periodic component in multiple climate series from the Arctic and the equatorial Pacific. It was determined that the signals shared a common source and a linking mechanism was found. The same techniques were applied for signal enhancement in ground penetrating radar.

The proposed wide spread influence of decadal solar variability on climate was scrutinized and it was concluded that the 11-year cycle sometimes seen in climate proxy records is unlikely to be driven by solar forcing.

Global sea level was reconstructed from tide gauge data using a new ‘virtual station’ method. The 1920-1945 rate of sea level rise was as large as the rate observed during the 1990s. The impact of major volcanic eruptions on global sea level was studied and it was found that a disturbance of the global water cycle causes a rise in sea level in the first year following an eruption.

Keywords: wavelet coherence, singular spectrum analysis, non-linear trend, phase-aware teleconnections, sea level, paleoclimate, ice core, continentality, Antarctic blue ice, glacier modelling

Acknowledgements

The research for this thesis was conducted at the Arctic Centre, University of Lapland and the department of geophysics, University of Oulu.

I am most grateful to my supervisor Docent John Moore for his guidance and friendship. I also want to thank Prof. Sven-Erik Hjelt for being supportive of my pursuits into glaciological research and for being open to cooperation between Oulu and Lapland universities.

I want to express gratitude to all my co-authors, contributors, and colleagues, many of which I now count as friends. In particular, I want to thank Dr. Svetlana Jevrejeva, Anna Sinisalo, Kristiina Virkkunen, and Teija Kekonen.

To all the members of the 2003/2004 FINNARP and the 2002 Lomonosovfonna expeditions: thanks for all the great experiences.

Finally, my deepest gratitude belongs to my wife Suvikukka. You inspire me to be the best that I can.

The work was financially supported by the Thule Institute, Academy of Finland, the Maj & Tor Nessling Foundation and the Finnish Antarctic Research Programme (FINNARP).

List of original papers

The following papers comprise the work in this thesis:

I Grinsted A, Moore J, Spikes VB & Sinisalo A (2003) Dating Antarctic blue ice areas using a novel ice flow model. Geophys Res Lett 30(19). doi:10.1029/2003GL017957.

II Grinsted A, Moore JC & Jevrejeva S (2004) Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlin Process Geophys 11: 561-566. (http://www.pol.ac.uk/home/research/waveletcoherence/).

III Grinsted A, Moore JC, Pohjola V, Martma T & Isaksson E (2006a) Svalbard summer melting, continentality, and sea ice extent from the Lomonosovfonna ice core. J Geophys Res 111: D07110. doi:10.1029/2005JD006494.

IV Grinsted A, Moore JC & Jevrejeva S (2006b) Observational evidence for volcanic impact on sea level and the global water cycle. (Manuscript).

V Moore & Grinsted (2006) Singular spectrum analysis and envelope detection: methods of enhancing the utility of ground penetrating radar data. J Glaciol 52(176): 159-163(5).

VI Moore J, Grinsted A & Jevrejeva S (2006a) Is there evidence for sunspot forcing of climate at multi-year and decadal periods? Geophys Res Lett 33: L17705. doi:10.1029/2006GL026501.

VII Jevrejeva S, Moore JC & Grinsted A (2004) Oceanic and atmospheric transport of multi-year ENSO signatures to the polar regions. Geophys Res Lett 31: L24210. doi:10.1029/2004GL020871.

VIII Jevrejeva S, Grinsted A, Moore JC & Holgate S (2006c) Nonlinear trends and multiyear cycles in sea level records. J Geophys Res 111: C09012. doi:10.1029/2005JC003229.

In the text these papers are referred to by authors and year followed by the roman numerals in superscript e.g. “Grinsted et al. (2006aIII )”.

In papers V, VI, VII, and VIII, A. Grinsted was responsible for the development of the mathematical methods and their practical application, while the first author and second authors were responsible for the interpretation of the results.

Contents

Abstract Acknowledgements List of original papers Contents 1 Overview ....................................................................................................................... 11 2 Time series analysis....................................................................................................... 13

2.1 Introduction ............................................................................................................13 2.1.1 Regression analysis .........................................................................................14 2.1.2 Stationarity ......................................................................................................14 2.1.3 Non-linear systems ..........................................................................................14 2.1.4 Time domain versus Frequency domain ..........................................................15 2.1.5 Statistical significance .....................................................................................16 2.1.6 Method of delays .............................................................................................17 2.1.7 Singular Spectrum Analysis (SSA)..................................................................17 2.1.8 SSA Non-linear trend.......................................................................................19

2.2 Exploring linkages..................................................................................................21 2.2.1 Frequency domain methods.............................................................................21 2.2.2 Time-Frequency domain methods ...................................................................22 2.2.3 Teleconnection patterns ...................................................................................24 2.2.4 EOF analysis....................................................................................................25 2.2.5 Instantaneous phase .........................................................................................25 2.2.6 Phase coherence...............................................................................................26

2.3 Global sea level ......................................................................................................27 2.3.1 Contributions to sea level rise .........................................................................28 2.3.2 Reconstructing GSL from tide gauges.............................................................29

3 Glaciological modelling ................................................................................................ 31 3.1 Ice core proxies.......................................................................................................31

3.1.1 Svalbard...........................................................................................................33 3.1.2 The Lomonosovfonna 1997 ice core ...............................................................33

3.2 Ice flow modelling..................................................................................................34 3.2.1 Antarctic blue ice areas....................................................................................35

4 Objectives of the thesis.................................................................................................. 37 5 Results ........................................................................................................................... 38

5.1 Dating Antarctic blue ice areas (Paper I) ................................................................38 5.2 Wavelet coherence (Paper II, Paper VI)..................................................................40 5.3 Lomonosovfonna Melt and Continentality (Paper III) ...........................................41 5.4 Sea level (Paper IV & VIII) ....................................................................................41 5.5 Noise reduction in Ground Penetrating Radar (Paper V) .......................................43 5.6 ENSO-Arctic links (Paper VII) ..............................................................................43 5.7 Sunspots and climate (Paper VI) ............................................................................43

References Original papers

1 Overview

Public interest into climate change research is increasing as people see the local impacts of anthropogenic global warming (e.g. ACIA 2004). Projecting the consequences of different courses of action (IPCC 2001a, ACIA 2004) is only one aspect of climate change research and relies on thorough understanding of the Earth climate system. Past behaviour can give clues and insights into the inner workings of the system and can serve as both calibration and validation data for the projection models. The past is also the context in which recent changes should be seen. Unfortunately systematic weather monitoring is very sparse prior to 1950 and virtually non-existent pre-1850. Proxy climate indicators from such sources as ice cores, phenology, and tree rings are very valuable. E.g. using proxy climate reconstructions there is evidence that post-1990 Northern Hemisphere (NH) mean temperatures are higher than they have been for the last 2000 years (Moberg et al. 2005).

At present there is a great dichotomy of opinion concerning the behaviour of the climate. In general, the modelling community sees climate from a deterministic world view where the present reality is considered to be a direct consequence of the past. In contrast, some mathematicians consider the climate to be a manifestation of stochastic scaling processes (e.g. Koutsoyiannis 2005), whereby even events such as the glacial/inter-glacial cycles may be considered ‘random fluctuations’. The approach taken to climate here is to use the tools and concepts from both camps to interpret past climate records in terms of plausible hypotheses on the mechanisms of climate change.

Increases in computing power have allowed more focussed regional scale climate modelling to be done. Therefore climate proxies from the central parts of Greenland and Antarctica are no longer sufficient. There is a need for proxies from wider and more typical regions than just the large ice sheets. This is especially important in remote areas such the Arctic where instrumental data and historical evidence is limited. Advances in sea ice modelling have led to the need for better records and proxies of past sea ice extent. The distance to open water from a site can be considerably modified by the presence of sea ice. This effectively causes a more continental climate locally with larger seasonal temperature range. We can therefore hope to reconstruct past sea ice extent through a continentality proxy.

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Antarctic blue ice areas have ancient ice exposed on the surface and are usually located much nearer to the coast than the traditional deep ice cores (Bintanja 1999). This is a valuable source of paleoclimatic data on the southern ocean that has not been utilized to date, largely because of difficulties in dating the ice. As with vertical ice cores, flow modelling is one approach to dating the ice. However, most BIAs are found in proximity with mountain ranges and nunataks which complicate flow modelling considerably (Bintanja 1999).

Global sea level rise (GSL) is a critical issue for billions of people on the planet. Indeed it can be argued that this is the single largest threat from anthropogenic global warming – both through the direct loss of available land area and through increased penetration of coastal flooding to inland areas. To assess the performance of models and to detect any acceleration in sea level rise, it is necessary to establish a good record of past GSL. Satellite altimetry provides the only direct measurement of sea level globally, but the altimetry data begins only in 1992. Tide gauges provide measurements of relative sea level at points located along the coast and we can reconstruct GSL back to ~1850 using these measurements. In practice, this is complicated by the time-varying spatial distribution of tide gauge records and vertical land movement of the station reference. The contributions to sea level rise can be divided into two major groups: Contributions to the total mass of the oceans and changes in the average density (due to heat content). It is important to assess the relative importance of these contributions on different time scales. Comparisons between observed and modelled regional sea level may provide answers in this respect.

2 Time series analysis

2.1 Introduction

Climate change implies a change over time and the tools from time series analysis are therefore natural choices when researching this subject. Time series analysis does not comprise a particular set of tools and is only defined by the subject of the analysis. A time series is a finite series of successive observations usually taken at regular intervals. In geophysics we encounter many such time series. Some examples are the yearly record of maximum ice extent in the Baltic Sea, monthly global sea surface temperature, or the hourly sea level at a tide gauge. Using time series analysis we can attempt to characterize, predict, and model the system. E.g. we may examine how time series from different parts of the system are linked, and in this manner gain insights into the mechanisms involved.

As physicists we may imagine the Earth climate system to be a deterministic system for which it, in theory, should be possible to write down the governing equations. We can describe the state of the system by a set of numbers such as the pressure at every point in the atmosphere. This set of values describes the state of the system completely and can be called a coordinate in the phase (/state) space of the system.

It is clear that the phase space of the climate system must be very high dimensional. However, pressure and temperature are useful concepts when dealing with the motion of billions of molecules and the laws of statistical mechanics justify the use of macro scale physics. Hence, we can hope to describe the interesting dynamical behaviour of the system in a reduced phase space with only a few active dimensions. Analyzing past behaviour of a complex system is notably different from the operational weather forecasting problem where sensitivity to initial conditions limits utility. Time series traditionally attributed to linear systems with many degrees of freedom (d.o.f.) can be generated by non-linear systems with few d.o.f. (Hilborn 2000). We can use time series analysis to uncover a pseudo-basis for the active dimensions and create a skeleton of the system dynamics in this reduced space.

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2.1.1 Regression analysis

The simplest linear systems respond passively to an external forcing. Regression analysis is in these cases very useful for finding the relationship between the forcing and the response. Many systems, however, have an internal memory. The response depends both on the forcing and on the past observations of the system (e.g. a forced oscillator). Regression analysis has been generalized for such systems in e.g. auto-regressive/moving average (ARMA) analysis (Sprott 2003). These systems are very well understood, but appear to be of very limited value in real geophysical applications where non-linearities are significant (see also section 2.1.3 ). They are, however, valuable as they provide a simple null-hypothesis to test the significance of features, such as quasi-periodic oscillations or trends, against. This is further discussed in section 2.1.5 .

2.1.2 Stationarity

Traditional time series analysis methods generally implicitly assume that the data come from a linear dynamical system plus noise. For such systems the variability in the series can be described by a superposition of basis functions such as exponentials and sines (Sprott 2003). Obvious examples are linear trend analysis and Fourier analysis. If the signal appears complex then a high order linear model is needed to capture its variability. The value of the model can be gauged by its utility if we were to repeat the measurement. This is closely connected to the stationarity of the underlying process (Kantz & Schreiber 2004). We call a process stationary if the probability density functions (p.d.f.) of all the relevant parameters for the dynamics remain constant through time (Kantz & Schreiber 2004). Unfortunately, what is relevant change from application to application. There is no absolute test for stationarity. Instead, we may test if the observed time series show non-stationary behaviour. As a first characterization of a time series we often look at the p.d.f. and a common definition of weak stationarity is that the mean and the variance (and auto covariance) does not change over time. In reality, we often have no possibility to re-measure the time series and must instead rely on comparing the statistical properties in different windows of the time series. Following the definition for weak stationarity the dynamical behaviour of e.g. linear growth would be a non-stationary process because the mean is changing over time (Sprott 2003). In the following I will use the word stationary more loosely in the spirit of the original meaning where relevance also has importance.

2.1.3 Non-linear systems

The deterministic world view is a fundamental aspect of science which largely relies on the repeatability of experiments. It was therefore a great surprise when simple deterministic systems showed behaviour that was neither repeatable nor predictable (Lorenz 1963). A tiny change in initial conditions or a slightly different external forcing

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produces wildly different outcomes even when the governing equations are known exactly. This type of behaviour has come to be known as chaos.

Chaos only occurs in systems whose governing equations are non-linear. We call such systems non-linear systems. Not all non-linear systems exhibit chaotic behaviour and some exhibit it to such a degree that it is practically indistinguishable from random noise. Linear time series analysis is applicable if the system is linear or if the chaotic behaviour can be approximated by random noise (Gershenfeld 1999). The reality is usually somewhere in between, and this is where non-linearity matters.

Natural time series such as those observed in the climate system are most likely generated by open Hamiltonian systems where energy is gained externally and dissipated internally (Lorenz 1963). The invariant sets towards which all solutions tend towards asymptotically are called strange attractors (Ruelle & Takens 1971). A long time series from a forced dissipative system can be used to reconstruct the attractor of the system because the solution of the system will densely map out the attractor in a state space of appropriate dimension. However, in practice the presence of noise and limited number of observations in almost all geophysical time series limits the usefulness of this idea.

When we consider the range of dynamical systems from the simple to the complex we find in turn: stable equilibria (fixed points), stable periodic solutions (or limit cycles), quasi-periodic solutions lying on tori, and strange attractors (Ghil et al. 2002). On this road to chaos we find fixed points and limit cycles. These simple attractors can still provide meaningful analogues for the system behaviour in some parts of phase space even though they may be unstable in some directions (some of their local Lyaponov exponents are positive). Such partially unstable attractors are called ‘ghost fixed points’ and ‘ghost limit cycles’ (Ghil et al. 2002). When the system gets close to a ghost fixed point then it will tend to linger in the neighbourhood for a time. Similarly, when the system gets close to a ghost limit cycle then it will likely follow a few orbits around the cycle. This will lead to quasi periodic behaviour as long as the system is near the ghost limit cycle.

2.1.4 Time domain versus Frequency domain

Time series do not have to be analyzed in the time domain. The observations can also be presented in other forms without loss of information. It is common to inspect the data in the frequency domain by e.g. Fourier transforming the data. The quasi-periodic behaviour of ghost limit cycles will produce peaks in the power spectral density in the frequency domain. One disadvantage of frequency domain methods is that they implicitly assume that the signal is stationary in time. Ghost limit cycles are inherently unstable and will only lead to near periodic behaviour as long as the system is in that region of phase space.

Many time domain methods also assume stationarity. To overcome the problem of non-stationarities it is common to apply the methods on sub sections or moving windows of the data. E.g. Moving averages of a time series can be used to detect regime shifts where the statistical properties change abruptly.

We can also present the data in a hybrid time-frequency domain by making spectral analysis in a moving window. This type of analysis is called Windowed Fourier

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Transform (WFT). In a window of a certain size will sample more oscillations of at short wavelengths compared to long. The WFT features will therefore be more robust at short wavelengths. Further for short windows aliasing of high and low frequency components become a serious issue. The WFT represents an inaccurate and inefficient method of time-frequency localization (Torrence & Compo 1998). A more useful method of presenting the data in time-frequency space is wavelet transforms. There are two classes of wavelet transforms; the Continuous Wavelet Transform (CWT) and its discrete counterpart (DWT). The DWT is a compact representation of the data and is particularly useful for noise reduction and data compression whereas the CWT is better for intuitive feature extraction purposes.

A wavelet is a function with zero mean and that is localized in both frequency and time (Torrence & Compo 1998, Foufoula-Georgiou & Kumar 1995). The idea behind the CWT is to apply the wavelet as a series of band pass filters to the time series. The centre frequency of the band pass filters is continuously altered by stretching in the wavelet in time and normalizing it to have unit energy.

2.1.5 Statistical significance

In order not to over-interpret the information in a time series it is important to assess the statistical significance of the results. The significance is estimated by calculating the probability that the same or an even more conclusive result may have occurred by chance. In order to do that it is necessary to define what is meant by chance. This is called the null-hypothesis for the statistical test.

In many traditional methods the null hypothesis is that the signal is generated by an uncorrelated Gaussian noise process. This is called a white noise null-hypothesis because the power spectrum is flat for such a process. Geophysical time series, however, have distinctive red noise characteristics with much more variability on long wavelengths than on short. This usually reflects the fact that there is some memory in the system. The observed value at any given time is not independent from the previous observations and therefore the effective d.o.f. is usually much smaller than the number of observed values in the time series. This means that traditional methods such as correlation analysis over-estimate the significance unless this reduction in the d.o.f. is taken into account. An auto-regressive noise model of order 1 (AR1) is the simplest and most common model for red noise and this is a much better choice for the null-hypothesis than pure white noise (with an AR1 coefficient of zero). The theoretical background spectrum is usually not known and the AR1 coefficient used in the red noise null-hypothesis therefore has to be estimated from the observed series (Allen & Smith 1996).

In practice, many time series methods do not have analytical expressions for how to estimate significance against a red-noise null-hypothesis. Fortunately increased computer power allows us to simply generate many surrogate noise data series with the proposed noise characteristics and run them through our time series analysis tools. We can then easily calculate the empirical probability distribution function for the measure of interest, and hence the significance. This approach is called Monte Carlo significance testing. One of the strengths of Monte Carlo testing is that it is easy to adapt the null-hypothesis to a

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particular case. It is e.g. a simple matter to generate noise with exactly the same Fourier power spectrum as the observed series (Gershenfeld 1999).

2.1.6 Method of delays

The state of an N-dimensional deterministic system can be uniquely characterized by N independent variables. The method of delays refers to a technique that enables us to sketch out a basis for the state space from a univariate time series measured in the system. The ‘natural’ basis spanning the state space is only defined within a specific mathematical formulation of the system and can not be retrieved from the single time series. However, it is useful to think of the measured time series x={x1, x2, … , xn} as the output of a measurement function xi=h(yi) that maps the state described by the vector yi into a scalar xi. The method of delays maps a univariate time series through a delay coordinate function H,

( )1 1( ) , , ,i i m i iH x x x x− + −= , (1)

where m is called the embedding dimension or lag. Mapping the entire time series x through H(x) will construct a trajectory matrix X with m columns and n-m+1 rows. Takens’ theorem (Takens 1981) guarantees that any smooth attractor can be reconstructed using X with under certain conditions. The mathematical conditions are rarely, if ever, strictly fulfilled for geophysical time series. The concept of a measurement function is problematic for several reasons. Climate time series will contain noise and thus can not be said to be the output of a measurement function of the ‘climatic state’. Further, numeric truncation errors further violates the assumption that the measurement function has a continuous first derivative. Despite this, the theorem still gives hope that we can learn as much from a trajectory in the delay coordinate system, as from a trajectory in the physically meaningful phase space.

2.1.7 Singular Spectrum Analysis (SSA)

Geophysical time series will often be highly auto correlated and the delay coordinates will hence be highly correlated. SSA extends the method of delays by linearly transforming the delay coordinates into a coordinate system where the covariance matrix is diagonal. In this respect SSA is very similar to EOF analysis and Principal Component Analysis. Further, the method successively maximizes the variance explained by the components. In this new coordinate system the individual components of the lagged time series are linearly independent. Hence, it is often reasonable to assume that the components represent signals generated by different physical mechanisms. A recommendable introduction to SSA applied to geophysical time series can be found in Ghil et al. (2002).

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A fundamental step in SSA is to choose the embedding dimension m. Takens’ theorem states that a lag more than twice the attractor dimension will be enough to hold the attractor. However, usually we do not know the attractor dimension and we find that dimensionality measures from chaotic time series analysis are not really useful in the context of SSA as the time series usually have quite limited temporal coverage. Usually the embedding dimension is chosen on the basis of what periodicities are of interest. Ghil et al. (2002) advice: “The choice of m is based on a trade-off between two considerations: quantity of information extracted versus the degree of statistical confidence in that information. The former requires as wide a window as possible, i.e., a large m, while the latter requires as many repetitions of the features of interest as possible, i.e., as large a ratio n/m as possible.” As a rule of thumb m is chosen so that n/m>5. Vautard et al. (1992) suggest that SSA is typically successful at analyzing periods in the range from m/5 to m. Variability on longer periods will usually be represented by a single component called the ‘non-linear trend’.

In order to find the projection from the delay coordinate system to the system where the variation in the principal directions are linearly independent we must first estimate the covariance matrix (C) of the trajectory matrix X (where x and X are defined in the previous section). Usually the mean is removed from x prior to its mapping through H, as this tends to give more independent columns in X. C is then diagonalized using eigenvalue decomposition. We write

TE CE = Λ , (2)

where E is a m×m unitary matrix of eigenvectors, Λ is a diagonal matrix of eigenvalues sorted by magnitude, and T denotes a transpose. The eigenvectors (or the columns in E) can be interpreted as temporal patterns in x and are called the EOFs in parallel to the spatial patterns from EOF analysis (see also section 2.2.4 ). We can interpret these temporal patterns as the dynamical signature left behind by orbits close to ghost limit cycles. In a sense the EOFs describe a skeleton of the underlying attractor. We project X onto the basis spanned by E,

A XE= , (3)

where the columns in A are called the principal components (PCs). Note how a column in A also can be calculated as the convolution of x with the corresponding eigenvector in E (in reverse order). The eigenvalues tells the amount of (co)variance captured by each PC. The leading EOFs (those with highest eigenvalues) will capture the strongest temporal patterns. Noise will be distributed among all the components. SSA will always return as many components as the embedding dimension, even if the temporal structure of the signal is very simple and can be captured completely by a few components. It is therefore important to assess which SSA components are significant. In Monte Carlo SSA (MC-SSA) we check if the amount of variance (i.e. Λ) is greater than what we would expect by chance from projecting red noise onto the EOFs (Ghil et al. 2002).

We can reconstruct the part of x from associated with a given EOFs. We write the ith Reconstructed Component (RC) as

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1 *ti i iMR A E= , (4)

where subscript i denotes the ith column, * denotes convolution, and Mt is the number points of overlap in the convolution at any given time. The sum of all the RCs give exactly x. We can also make a partial reconstruction where we discard e.g. noise components from the sum (Moore & Grinsted 2006V ). The ith RC can also be calculated as Mt

-1x*Êi*Ei, where ^ means in flipped in reverse order. In the central part of the time series Mt=m and we can consider m-1Êi*Ei as a Finite Impulse Response (FIR) filter that we are applying to x. This FIR filter will always be symmetric and hence have a linear phase response.

2.1.8 SSA Non-linear trend

“Proper smoothing of climate time series, particularly those exhibiting non-stationary behavior (e.g., substantial trends late in the series) is essential for placing recent trends in the context of past variability.” (Mann 2004). In SSA, low-frequency variability with periodicities longer than the embedding dimension is usually captured by a single component. In the literature this component has been dubbed the nonlinear trend (Ghil et al. 2002). The RC representing the trend can be considered to be the output of applying a data adaptive low pass FIR filter to x. The advantage of using this SSA derived FIR filter rather than traditional low pass filters is that leading oscillatory components will be accounted for prior to the trend detection. Below I will develop a new method of treating the boundaries as well as a new method of obtaining confidence intervals for the non-linear trend. I will use annual ‘Global’ Sea Surface Temperature (SST) in the latitude band 60°S-60°N calculated from ERSSTv2 (Smith & Reynolds 2004) as an example.

Fig. 1. Reconstructing FIR filter corresponding to the SSA (m=15) leading component of the global SST (60°S-60°N).

Lag (years)

Coe

ffici

ent

−30 −20 −10 0 10 20 300

0.005

0.01

0.015

0.02

0.025

0.03

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The FIR filter will reach beyond the boundaries of the time series, and it is important to consider which boundary conditions should be employed. Two useful boundary conditions are the minimum slope and the minimum roughness conditions (Mann 2004). The minimum slope condition minimizes the first derivative, and minimum roughness minimizes the second derivative at the boundary. Mann (2004) implements the minimum roughness condition by padding the series with the time series mirrored around the end point (both horizontally and vertically), and the minimum slope by mirroring horizontally. Here, I will use a variation of the minimum roughness criterion where the series is padded with a linear extrapolation based on the m preceding points.

Fig. 2. The global SST anomaly (60°S-60°N) (dotted line). The SSA non linear trend (m=15, min. roughness) and its 2σ confidence interval is shown as a solid line and grey shading. The confidence interval is calculated from the standard errors reported for the ERSSTv2 data set.

We can estimate a confidence interval around the smoothed time series by considering the effect the FIR filter has on noise with the same characteristics (i.e. variance and colour) as the errors. In practice, this can easily be accomplished through Monte Carlo methods. The FIR filter is applied to a large surrogate set of noise series and the confidence interval is estimated from the smoothed noise distributions. We can also quantify the errors associated with the extrapolations beyond the boundary, by e.g. measuring the misfit when extrapolating an m-point linear trend inside the time series. This method of obtaining confidence intervals is also applicable to traditional low pass filters. In Fig. 2 I show the SSA non-linear trend and its confidence interval of the global SST.

Year (A.D.)

SS

T A

nom

aly

(°C

)

1860 1880 1900 1920 1940 1960 1980 2000

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

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2.2 Exploring linkages

Physical relationships between components of the climate system will be manifested in the time series we observe. Hence, we may attempt to gain insights into the underlying mechanisms by finding evidence for relationships in the observed time series. Traditionally, the similarity between time series has been measured through correlation coefficients. Similarity may, however, be caused by different things. Perhaps the similarity is a consequence of the time series being subject to the same external forcing. This is a case that occurs quite often when correlating climate series that span the industrial period. Almost all parts of the system have a clear trend due to anthropogenic global warming. Other times the similarity may be better explained by one series forcing the other. Estimated leads and lags between the signals recorded in the time series are valuable evidence for the mechanisms involved.

2.2.1 Frequency domain methods

Another tool that is often used to explore links is comparing the power spectra of the two series (e.g. Rigozo et al. 2002). However, the reasoning that a link should exist simply because of a similarity of a particular spectral peak is weak for several reasons. Most series have several spectral peaks it is therefore relatively likely that peaks would occur at the nearby frequencies. Secondly, a slight offset in the fundamental period would cause the oscillations to move in and out of phase. Therefore it is very important to assess whether the two series have a consistent phase relationship.

The similarity between 2 power spectra can be objectively assessed through Cross Spectral Density (CSD) estimates and the significance can be tested against red noise using Monte Carlo methods. The CSD of the time series X and Y is defined as the magnitude (complex modulus) of

*( ) ( ) ( )XY X YC f F f F f= ⋅ , (5)

where F is the Fourier transform, f is the frequency and * denotes a complex conjugate. The complex argument of CXY can be interpreted as the average phase difference between the series. However, the CSD only tell how much common power there is on a given wavelength and does require a consistent phase relationship. The magnitude squared coherence (R2) is a measure of the consistency of the phase relationship and is calculated as

2

22 2

( )( )

( ) ( )

XYw

XYX Y

w w

C fR f

F f F f=

⋅

∑

∑ ∑, (6)

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where the ∑w indicates that the terms should be summed over a series of windows. The coherence varies between zero and one, taking on the value 1 when the complex argument of CXY is constant for all the windows. However, just as with the windowed Fourier transform there are several drawbacks due to the windowing involved (see section 2.1.4 ).

2.2.2 Time-Frequency domain methods

The ideas of the cross spectral density and the magnitude squared coherence has been applied in the time-frequency domain through continuous wavelet methods. The definition for the Cross Wavelet Transform (XWT) closely follows the definition for the Cross Spectral Density (Torrence & Compo 1998, Grinsted et al. 2004II ). Assuming that a complex wavelet is used then the complex argument of the XWT can be thought of as the instantaneous relative phase at that wavelength. Parallel to the caveat with the CSD, peaks in XWT only indicate high coincident power and not a consistent phase relationship. Maraun & Kurths (2004) goes as far as stating “The wavelet cross spectrum appears to be not suitable for significance testing the interrelation between two processes”. It is therefore useful to overlay plots of XWT with phase arrows, so that the phase relationship can be readily examined by eye. A significance test against an AR1 null hypothesis for the cross wavelet power can be found in Torrence & Compo (1998).

Wavelet Coherence (WTC) is the wavelet analogy of coherence in Fourier domain. The definition is very similar to eq. 5, with CXY, FX, and FY replaced by their wavelet counterparts and with the sums replaced by a smoothing operator (Torrence & Webster 1999). The significance of the WTC against red noise is done through Monte Carlo methods. However, we find that the significance level is close to independent of the noise colour (Grinsted et al. 2004II ). This is because the expression for the WTC is normalized with respect to the local power under the smoothing operator. I have developed an open source wavelet coherence toolbox for Matlab, which is freely available on the web (http://www.pol.ac.uk/home/research/waveletcoherence/).

23

Fig. 3. Wavelet squared coherence between the standardized AO and BMI time series. The 5% significance level against red noise is shown as a thick contour. The relative phase relationship is shown as arrows (with in-phase pointing right, anti-phase pointing left, and BMI leading AO by 90º pointing straight down). The cone of influence (COI) where edge effects might distort the picture is shown as a thick black line. All significant sections show anti-phase behaviour.

An example of two physical systems that we expect to be linked from consideration of the climate system are the mean winter state of the arctic atmosphere and winter severity reflected by ice conditions. Here I will analyze winter AO (Thompson & Wallace 1998) and maximum annual ice extent in the Baltic Sea (BMI; Seinä et al. 2001) with special attention to the phase relationships between the series in the light of the expected causality links. Recently published results demonstrate that large-scale atmospheric circulation patterns in the Arctic and North Atlantic described by the AO or by the somewhat similar North Atlantic Oscillation teleconnections significantly control ice conditions in the Baltic Sea (Loewe & Koslowski 1998, Omstedt & Chen 2001, Jevrejeva & Moore 2001, Jevrejeva 2002).

The squared WTC of AO and BMI are shown in Fig. 3. There is strong coherence with clear anti-phase behaviour in almost all regions of time-frequency space. Oscillations in AO are manifested in the BMI on wavelengths varying from 2-20 years, suggesting that BMI passively mirrors AO. Regions with low coherence coincide with low wavelet power in the AO and are therefore expected. The reason for the low power in of AO pre-1900 could be because the associated pressure pattern is not capturing the actual location of the centres of action during the Little Ice Age.

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2.2.3 Teleconnection patterns

In the 18th century Hans Egede Saabye, then a missionary in Greenland, wrote in his diary: “In Greenland, all winters are severe, yet they are not alike. The Danes have noticed that when the winter in Denmark was severe, as we perceive it, the winter in Greenland in its manner was mild, and conversely.” (Saabye 1942). This statement strongly suggests that what is now known as the North Atlantic Oscillation (NAO) was common knowledge at the time (see also Crantz 1765). In the book “The Kings Mirror” circa 1230 A.D., there is even some evidence that the Norse knew of this phenomenon (Stephenson et al. 2003). Such observations of systematic behaviour can give valuable insights into the mechanics of the climate system.

Fig. 4. South Greenland (marked with a black dot) Sea Level Pressure (SLP) teleconnection pattern. Prior to analysis the time series at each grid point has been linearly detrended. Notice the anti-correlation between the SLP in Greenland and that in Western Europe and the tropical North Atlantic.

Angström (1935) used ‘teleconnection patterns’ to examine the NAO see-saw. Teleconnection patterns are a quantitative method of visualizing the spatial structure of links between a gridded dataset and the observations at a given location. Traditionally the strength of the links is estimated through correlation coefficients. The term has since then been generalized so that it also encompasses correlation maps between two completely different time series. For example Bjerknes (1969) demonstrated links between a Southern Oscillation Index (SOI) derived from Sea Level Pressure (SLP) and tropical pacific sea surface temperature (SST). Based on his results he proposed a two-way coupling between ocean and atmosphere. The theory behind this coupling has since matured and is now known as the El Niño-Southern Oscillation (ENSO) (see Battisti &

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Sarachik 1995 for a review). In Fig. 4 I have calculated the South Greenland teleconnection pattern in detrended SLP. The figure was made using the HADSLP2 dataset down-sampled to yearly resolution (Allan & Ansell 2006).

2.2.4 EOF analysis

Teleconnection patterns can reveal large-scale spatial climatic links. However, the climatic patterns can hardly be called fundamental as the exact structure will depend on the location of the comparison site. Empirical Orthogonal Function (EOF) analysis was developed by Lorentz (1956), as an objective method for finding the most common spatial patterns. Mathematically, the procedure is equivalent to Principal Component Analysis (PCA) which has its roots in Hotelling (1933). It is common within the geophysical community to use the term EOF for gridded data sets and PCA otherwise (e.g. Moore et al. 2005a).

We construct an n×m matrix (X) consisting of the values (xij) in the jth grid cell at the ith time instant. Below I will use the HADSLP2 data set as an example, and the X values are in this case the area weighted monthly anomalies of SLP. The ith row of X will be an m-dimensional vector in what I will call grid space. The objective of EOF analysis is to find a linear transformation from grid space into an orthogonal space so that there is zero covariance between the columns. We write

A XE= , (7)

where E is the n×n matrix of eigenvectors in C sorted by their corresponding eigenvalues (λi) with the highest first. The variance explained by component(/column) number i of A is λi. Each yij will be a linear combination of the X values at the jth time instant with the coefficients ith row in E. For each component there is thus a set of coefficients to each grid cell. We can therefore make a map of these coefficients which we usually call the EOF pattern. The matrix E is called the eigenfunction matrix and in PCA terminology contains the loadings or coefficients (Sprott 2003).

It is common to use EOF analysis to create a reduced state space while minimizing the mean square residuals of the truncated representation. This can simply be done by eliminating the rows of E corresponding to the lowest eigenvalues. In practice C is often close to singular and a truncated version of E can be determined efficiently using singular value decomposition (Björnsson & Venegas 1997).

2.2.5 Instantaneous phase

Consider a time series generated by a simple harmonic oscillator (i.e. a simple sine wave). The space spanned by the time series and its numerical derivative (or alternatively a lagged version of the time series) can be considered a state space representation of the system and a trajectory in this space will form an ellipse. We can also describe the state using polar coordinates relative to the centre of this ellipse. The angle can then be

26

considered the instantaneous phase and the magnitude can be considered the oscillation strength or amplitude.

In geophysics we rarely find time series that are perfectly periodic. We can, however, expand the concept of instantaneous phase to signals that are quasi periodic. In this case we will be looking for a truncated representation of the true state space in 2 dimensions where the trajectory circulates around a central point. It is not necessary that the amplitude or the angular velocity is constant for the concept of instantaneous phase to be useful.

One way of isolating a quasi-periodic component is to convolute the time series with a wavelet. The centre frequency of the wavelet should be chosen so that it matches the characteristic frequency of the quasi periodic signal under investigation. It makes sense to use a complex wavelet where the complex argument directly can be considered the instantaneous phase. Another common method for extracting quasi periodic components is Singular Spectrum Analysis (SSA) which yields an orthogonal set of data adaptive filters in the time domain (see section 2.1.7 ). Both methods effectively band-pass filter the time series. It is important to consider how wide a pass band is appropriate for the signal. That is, the choice of wavelet and SSA lag should be considered carefully. Just as with choosing a wavelet in the wavelet transform this choice is based on a trade-off between frequency and time localization. The term ‘instantaneous phase’ implies that time localization is the most important, and the width of the pass band should therefore be chosen as large as possible but still restrictive enough that the filtered signal does not contain multiple quasi periodic components. In practice, I recommend examining the truncated state space trajectory to ensure that it evolves around a central point. The broad band Paul wavelet has better time localization than the Morlet (Torrence & Compo 1998) and is in many cases a good choice.

2.2.6 Phase coherence

We often come across cases where we would expect that the state of one system is related to that of another. E.g. one system could be forcing the other or the two systems could be responding to the same external forcing. The responding system can only react to the forcing as it happens and the responding system will normally lag the forcing. Hence, we would expect two time series from such systems to have a relative lag. The instantaneous phase difference of the signals is interesting as this may be interpreted as a lag. We define the phase coherence as angle strength (Zar 1999, Grinsted et al. 2004II ) of the time series of phase differences (see Moore et al. 2006aVI ). This can be thought of as the degree to which the signals are phase-locked. Testing the significance of the phase coherence can be done using Monte Carlo methods against any noise hypothesis. The circular mean of the relative phase angle series (Zar 1999, Grinsted et al. 2004II ), can be translated into a time-lag if the angular velocities (or characteristic frequencies) are well defined.

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Fig. 5. a) Phase coherence between SOI and Niño3.4 over the 20th century for different lags. The instantaneous phase has been extracted using a Paul wavelet with a 5 year centre periodicity. Horizontal dotted lines indicate the significance estimated using Monte Carlo methods on surrogate series with the same Fourier power spectra as the input series. b) The mean relative phase between SOI and Niño3.4.

To illustrate how phase coherence can be utilized I will apply this technique on two El Niño-Southern Oscillation (ENSO) time series. Here, I use the monthly Southern Oscillation Index (SOI) (Ropelewski & Jones 1987), and the monthly Niño 3.4 SST index (Smith & Reynolds 2004), defined as the SST averaged over the eastern half of the tropical Pacific (5˚S–5˚N, 120˚–170˚W). Typical ENSO variability is in the 2-7 year band (e.g. Battisti & Sarachik 1995). I therefore derive the instantaneous phase by applying a complex Paul wavelet with 5 year centre periodicity to both series after removing the annual cycle. In Fig. 5a there is highest coherence for a lag of -3 months, indicating that SOI leads Niño3.4 by ~3 months as may be expected physically (Clarke et al. 2000, Jevrejeva et al. 2004VI). The relative phase of SOI and Niño 3.4 is -162° at zero lag (Fig. 5b), or 18° out of perfect anti-phase. At 5 year periodicity 18° also corresponds to a 3 months lead of SOI. Monte Carlo testing against white noise or red noise makes only slight difference to the significance levels in Fig. 5, with white noise significance levels being shifted down by 0.03 and white noise levels by 0.01. Hence, a red noise significance test is sufficiently robust.

2.3 Global sea level

Changes to Global Sea Level (GSL) will have a profound impact on human societies and coastal eco-systems (IPCC 2001a). The projected increase in GSL from 1990 to 2100 is

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in the range from 0.08 m to 0.88 m (IPCC 2001a), with most of the uncertainty associated with future emission scenarios. The combination of a projected 40cm sea level rise and increase in population in the coastal zone in the period from 1990 to 2080 will put an additional 88-241 million people at risk from flooding, depending on the level of adaptive measures taken (IPCC 2001b). Besides increased flood risk, rising sea level also has many other socio-economic consequences such as salt water intrusions into fresh groundwater, and accelerated coastal erosion. Ecosystems that are vulnerable to sea level rise include mangrove forests, coral reefs and salt marshes.

2.3.1 Contributions to sea level rise

Sea level is determined by the total volume of water in the world oceans. Volume changes are driven by changes in both total mass and average density. Global warming is causing both to increase. Increases in the Global Oceanic Heat Content (GOHC) causes an associated thermal expansion of the world oceans. The largest non-oceanic water reservoirs (the ice sheets, glaciers, ground water, and permafrost) are all sensitive to warming. Changes in the amount of water stored in these reservoirs directly influences GSL. In Fig. 6 estimates of the different contributions to sea level rise (1910-1990) are shown.

Fig. 6. Individual contributions to sea level rise from 1910-1990 (light gray) and the total of the components compared to observation based estimates of the GSL rise (dark gray). Boxes show estimates for the minimum and maximum values and centre-line does not necessarily indicate the best estimate (IPCC 2001a). Exceptions are the Greenland and Antarctic contributions which show best estimates for the recent (over a ~10 year period) mass balance and the 95% confidence interval (IPCC 2001a). The range of the total was calculated as the 95% confidence interval assuming that the component boxes indicate the 95% confidence interval. The Jevrejeva et al. (2006VIII ), and Church & White (2005) estimates were calculated as the 1910-1990 linear trend of the respective GSL reconstructions and confidence intervals calculated as in Church & White (2005).

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The recent contributions from ice sheets and glaciers in Fig. 6 were calculated by making a mass balance budget (IPCC 2001a). Direct measurements are usually very sparse and the estimated net balance of the large ice sheets rely on heavy extrapolation (Rignot & Thomas 2002). The modelled contributions from the ice sheets 1910-1990 are considerably lower than estimates of their recent mass balance (IPCC 2001a, Fig. 6). It is not clear if the reason for these differences is due to a recent acceleration in the contributions, model errors, or measurement extrapolation errors. An alternative method to estimate the mass balance is to weigh the ice sheets by measuring gravity changes from satellites. The GRACE project has done that since 2002 and the first estimates of the Antarctic and Greenland contributions are 0.4 ± 0.2 mm/yr (Velicogna & Wahr 2006) and 0.2 ± 0.1 mm/yr (Velicogna & Wahr 2005) respectively. These estimates for Greenland agrees with independent estimates over roughly the last decade (Krabill et al. 2004, Box et al. 2004, IPCC 2001a). It should be noted that GRACE has only been collecting data for a few years and the trends are not necessarily representative for the long term climatic trends. However, GRACE can be used to test the both mass balance models and the extrapolation methods used in traditional mass balance budget estimates. This hopefully leads to a reduction in the error bars on ice sheet contributions. Similarly, GRACE also shows promise in reducing the large error bars on the terrestrial storage term (Tapley et al. 2004).

Sea level rise is a relentless consequence of global warming. IPCC (2001a) states “If greenhouse gas concentrations were stabilised, sea level would nonetheless continue to rise for hundreds of years. After 500 years, sea level rise from thermal expansion may have reached only half of its eventual level, which models suggest may lie within ranges of 0.5 to 2.0 m and 1 to 4 m for CO2 levels of twice and four times pre-industrial, respectively.” and continues “Models project that a local annual-average warming of larger than 3°C sustained for millennia would lead to virtually a complete melting of the Greenland ice sheet.” It is therefore worrying that recent studies indicate that the response time of the large ice sheets may be substantially faster than previously thought (Zwally et al. 2005, Cullen & Steffen 2001). Melting of the entire Greenland ice sheet corresponds to a sea level change of ~7 m and Antarctica would correspond to ~61 m allowing for isostatic rebound (IPCC 2001a).

2.3.2 Reconstructing GSL from tide gauges

Surprisingly, measuring sea level is not as easy as measuring the level of water in a bath tub. There are many difficulties and this leads to controversy when discussing recent accelerations in sea level (e.g. since the 1990s). It is also very difficult to establish what fraction of the present day rise may be attributable to simple volume changes in sea level and how much is due to an increase in ocean mass, similarly there is also dispute as to how important anthropogenic factors are and how significant are natural cycles of sea level. To resolve these issues we need to use the direct observational records of sea level. GCM modeling of sea level rise is presently not possible as the models do not have good hydrological cycle flux constraints. It is relatively little-known that the sea level rise scenarios produced by the IPCC are largely based on GCM projections of global ocean

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heat content and parameterizations of mass flux from the land. Satellite altimetry now provides virtually real-time and global coverage (excluding high latitude regions – which are of course of great interest in mass flux to the oceans) of the present state of sea level, but the coverage extends back only to 1992. Beyond that we only can rely on the network of tide gauge stations, which suffer from two main limitations: being poorly distributed around the world, located at continental coastlines and on islands; and being attached to the land, which can move vertically. Paradoxically the sea level of the oceans given by satellite data are not immediately comparable with that measured by tide gauges as various coastal processes such as coastal wave propagations, storm surges, tectonic movement, coastal erosion, development, and floods can dramatically alter local coastal sea level measurements (IPCC 2001a).

Disturbances to the tide gauge or its platform can be accounted for by measuring the sea level relative to geodetic reference benchmarks. Such records are called Revised Local Reference (RLR) records and are suitable for reconstructing GSL. However, vertical land movement of RLR benchmarks complicates the use of tide gauge records for GSL reconstruction. In particular land is still rebounding due to the removal of the large ice load at the last deglaciation. E.g. regions that were under ice are now steadily emerging and other are compensating by submerging (Douglas 1995, Peltier 2001). Hence, we have to correct the RLR records for the ongoing Global Isostatic Adjustment (GIA) before they capture sea level. Any error in the GIA corrections will cause a corresponding error in the GSL trend. Most recent studies have relied on either models of post-glacial rebound or geological evidence for the GIA corrections. Geological evidence has the disadvantage that it measures past vertical land movement which is not necessarily the same as the present. The GIA models rely on assumptions of the Earths interior and a loading history. Any errors in these assumptions will create errors that are coherent on large spatial scales. Unfortunately this means that averaging many tide gauge records does not necessarily reduce the error from the GIA model. Other sources of vertical land movement such as tectonic activity (especially earth-quakes) and coastal subsidence in river deltas should also be taken into account. Most estimates of 20th century sea level rise specifically exclude records that may be disturbed by earth-quakes.

In Jevrejeva et al. (2005) we show a link between AO and European sea level, which is best explained by a redistribution of water driven by shifts in surface air pressure. There is no trend in SLP as the atmosphere is essentially conservative, but there are significant differences on decadal time scales between SLP at different positions on the earth’s surface – primarily as a result of long period fluctuations in large scale atmospheric pressure systems like the El Niño-Southern Oscillation (ENSO) or Arctic Oscillation (AO). These can produce decadal signals amounting to 30% of the variance in regional tide gauge records (Jevrejeva et al. 2005). We may reduce the signal caused by air pressure anomalies from the tide gauge records prior to a reconstruction of GSL with an inverse barometer correction (e.g. Grinsted et al. 2006bIV ).

There is no common reference level for the RLR records. The tide gauge records can therefore not simply be averaged, unless they span the same period. However, the trends for different records can directly be compared. E.g. Douglas (1997) calculated the linear trend of 24 RLR records from sites with small GIA corrections, and found trends of 1.8 ± 0.1 mm/yr. Another approach, which we have found useful, is to average the annual sea level rate of many records and then integrate this to get an average GSL curve.

3 Glaciological modelling

3.1 Ice core proxies

Glaciers and ice sheets form where the amount of snow which falls each year exceeds the amount which is lost through melt. As layer upon layer of snow accumulates, ancient snow gets buried. Hence, glaciers and ice sheets can be regarded as atmospheric sediment. Mixed within the snow there will be small amounts of impurities such as e.g. atmospheric aerosol, volcanic ash, and wind-blown soil dust. The snow layers are gradually transformed into solid ice as the load of the snow above increases and within it small pockets of air containing samples of the atmosphere at the time will be trapped. Deeper the ice layers will be getting thinner and spreading out as the load increases further. The ice flows under influence of gravity from the accumulation area to the area where there is net loss of ice (the ablation area) due to e.g. melting and calving of ice bergs.

Ice cores drilled in a glacier or an ice cap will contain progressively older layers of ice with depth. The deep ice cores from the ice sheets in Greenland and Antarctica can span hundreds of thousands of years, but the smaller ice caps of the Arctic mostly contain only Holocene ice and typically records span a few hundred years (Koerner 1997, Delmas 1992, Wolff 1990). The past precipitation preserved in the ice core holds many potential proxies of climate. One of the most well known is δ18O which is the relative deviation of the isotopic ratio 18O/16O from the ratio in standard sea water (Dansgaard 1964). Water molecules containing the heavy isotope of oxygen (18O) have a lower vapor pressure than those containing the light (18O). This leads to a fractionation causing a change in δ18O during evaporation and condensation. The realization that δ18O in precipitation was a proxy for temperature spawned the first ice core drilling for paleoclimate purposes at Camp Century, Northwest Greenland (Dansgaard et al. 1969). Since then, δ18O in precipitation has become a generally accepted proxy for condensation temperature in the atmosphere and is related to the surface temperature in modern snow (van Lipzig et al. 2002). However, the relationship between δ18O and surface temperature changes under different climate regimes as it also is affected by changes in e.g. moisture source temperature, seasonality of precipitation and moisture transport path (Paterson 1994).

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It is necessary to know the age of the ice layers in an ice core in order for it to be a useful record of past climate. In some cores the seasonal signal in δ18O is well preserved and we can count the annual layers and in this way date the ice. However, often the seasonal signal is not very clear due to diffusion, low accumulation or percolating melt water. In these cases, we can model the thinning of the ice with depth and in this way obtain a modeled dating (Paterson 1994). The model can be constrained by dating horizons where we know the ice to be a certain age. E.g. the snow from 1963 has elevated levels of radioactivity due to atmospheric nuclear test carried out in preceding years.

The impurities in ice have many different sources which generally fall into these groups: marine, terrestrial, anthropogenic, biogenic, and volcanic. Ions are among the most commonly studied constituents in ice as they contain clues on the source of the air at the drill site. Some sources lead to very specific ion signatures. E.g. The only known source of methanesulfonate (MSA) is from di-methyl sulphate (DMS) emissions due to oceanic biogenic activity (Legrand & Mayewski 1997).

The air bubbles trapped in the ice allow us to measure past concentrations of atmospheric gases. Of particular interest are past concentrations of greenhouse gases such as carbon dioxide and methane. Using this method it has been found that post industrial levels of CO2 and CH4 are roughly 30% and 130% higher respectively than at any other time during the last 650000 years (Siegentaler et al. 2005, Spahni et al. 2005). One practical difficulty with the use of records extracted from the trapped gases is that the gases will be slightly younger than the surrounding ice. This is because the snow is permeable and the gases are ventilated until the ice becomes so dense that the pores within the firn close-off. The pore close-off age will vary from site to site due to differences in e.g. accumulation rate and temperature.

The seasonal difference between summer and winter is of broad environmental significance. Continental interiors have larger seasonal temperature variations than maritime areas and the annual temperature range is often used as a measure of continentality. Ice cores from the interiors of large ice sheets such as Greenland or Antarctica are probably not very sensitive to the location of the sea ice margin – which determines the distance to open water and hence influences continentality – as it is far away from the drilling site. However, an ice core from an island near the seasonal sea ice margin may reflect the much larger relative variations in distance to open water via a continentality index. In ice cores where the seasonal cycle in δ18O is well preserved the amplitude is an obvious candidate as a proxy for continentality. Unfortunately the oxygen isotope record is often affected by diffusion on a scale comparable to the annual layer thickness.

Almost all ice cores in the Arctic outside of Greenland are influenced by seasonal melt. Melt water can refreeze on the surface, percolate into deeper layers or run off. If melt water refreezes in the surface layers then it has little impact on average ion concentrations and δ18O values. Percolation and run off affects ions much more strongly than isotopes because most ions have a very strong affinity for the water phase and even small amounts of melt water will have high concentrations of ions (Davies et al. 1982). Percolation however only redistributes the ions in the snow pack whereas they are lost by run-off. Iizuka et al. (2002) used the preferential removal of ions from particular layers as an indicator for seasonal melt on Austfonna, Svalbard, and found empirically that the magnesium sodium ratio was the best melt proxy.

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3.1.1 Svalbard

Svalbard is surrounded by the Arctic Ocean, Barents Sea and North Atlantic. It is located near the overturning point of the North Atlantic thermohaline circulation, a key component in the climate system (Broecker 1991). The thermohaline circulation brings warm waters from the south to the Arctic and is responsible for the relatively mild climate of the Svalbard archipelago considering its northern position. Svalbard is located at the southerly edge of the permanent Arctic sea ice and has open ocean to its west even during winter. Glaciers cover 60% of Svalbard and most of the glaciers in Svalbard are polythermal (a layer of cold-ice on top of the temperate ice layer), (Pälli 2003). Relatively little is known about the regional climatic conditions prior to 1911. However, records of Barents Sea ice extent since 1864 have been compiled from ship logbooks (Vinje 2001). The most prominent pattern is a decreasing trend in ice extent throughout the record.

Several ice cores have been drilled on Svalbard ice fields producing records less than 1000 years long. Groups from the former Soviet Union drilled two ice cores on Lomonosovfonna in 1976 (Gordiyenko et al. 1981, Vaikmaë 1990) and in 1982 (Zagorodnov et al. 1984) and five ice cores from other sites around Svalbard between 1975 and 1987 (Punning et al. 1980, Zagorodnov & Zotikov 1981, Vaikmaë et al. 1984, Kotlyakov 1985, Punning et al. 1985, Punning & Tyugu 1991, Tarussov 1992). The Japanese have also drilled several ice cores at other sites in Svalbard since 1987, (e.g., Goto-Azuma et al. 1995, Watanabe et al. 2001, Matoba et al. 2002). However, only a few of these cores have been studied in detail. The best studied core from Svalbard is the 1997 Lomonosovfonna core drilled and analyzed by a joint Dutch-Finnish-Norwegian-Swedish group (Isaksson et al. 2001).

3.1.2 The Lomonosovfonna 1997 ice core

In 1997 a 121 m long ice core was drilled (spanning about 800 years) on Lomonosovfonna (Isaksson et al. 2001), the highest ice field in Svalbard (78°51'53"N, 17°25'30"E, 1255 m a.s.l.). The site was chosen as published data from a previous, lower-elevation ice core on Lomonosovfonna drilled in 1976 indicated better preserved stratigraphy than from other sites on Svalbard (Gordiyenko et al. 1981). Total ice depth from radar sounding was 123 m, and the site is close to the highest point of the ice cap with roughly radial ice flow. Dating of the core was based on a layer thinning model tied with the known dates of prominent reference horizons (see Kekonen et al. 2005a for details).

Studies so far indicate that the Lomonosovfonna ice core, despite seasonal melt, contains a reliable record of isotope and chemical concentrations that can be successfully used in climate and environmental studies (e.g., Moore et al. 2005a, Isaksson et al. 2001). The soluble ion chemical record (Cl-, NO3

-, SO42-, CH3SO3

-, NH4+, K+, Ca2+, Mg2+, Na+)

in the ice core spans the last 800 years (Kekonen et al. 2005a) and shows clearly defined changes corresponding with different climatic periods and anthropogenic pollution. Isaksson et al. (2001) and Kekonen et al. (2002) reported increases in sulphate and nitrate

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in the mid twentieth century. Although local sources of pollution have to be taken into account (Simões & Zagorodnov 2001), there are also clear anthropogenic signals from long-range transport to the Lomonosovfonna ice cap (Kekonen et al. 2002), probably from both Eurasia and also from North America. Moore et al. (2006cXIII) made a sulphate inventory of the different sources of sulphate and found Western Europe to be the dominant anthropogenic source. Vehviläinen et al. (2002) found PAH compounds in the core and they concluded that these come from long-range sources. There is evidence that δ18O is a good proxy for surface air temperature at the drill site: the rise in δ18O from 1900 to 1920 that mirrors the rise in temperature seen in the instrumental record from Svalbard (Isaksson et al. 2003), consistency with the borehole temperature profile (van de Wal et al. 2002), and well preserved annual cycles (Pohjola et al. 2002b). Isaksson et al. (2003) suggested that the twentieth century was the warmest during the past 600 years based on δ18O record from both the Lomonosovfonna and also the Austfonna ice cores. O’Dwyer et al. (2000) discussed methanesulfonic acid variations in terms of Barents Sea conditions. The annual accumulation has been calculated back to 1715A.D. by Pohjola et al. (2002b). The borehole temperature profile is modeled by Van de Wal et al. (2002) and shows that the nineteenth century was about 2°–3°C cooler than the twentieth century. Melting in the Lomonosovfonna ice core was studied in detail by Pohjola et al. (2002a) and showed that nitrate and sulphate are most easily percolated, while ammonium and oxygen isotopes are virtually unaffected. About 80% of calcium in the core is of non sea salt origin, and its covariation with sulphur has been interpreted in terms of a terrestrial source by Kekonen et al. (2005a). The clear volcanic peak following the 1783 Laki eruption was discussed in detail in Kekonen et al. (2005b).

3.2 Ice flow modelling

The modelling of ice flow can be used for many different purposes. Flow models may be used to study the response of glaciers and ice sheets to changes in e.g. mass balance or temperature (e.g. Näslund et al. 2000). In the context of ice cores, flow models are generally used to derive the relationship between age and depth (e.g. Dansgaard & Johnsen 1969). In some cases we may have observational data that severely constrains the output of the flow model. In these cases we can use inversion methods to infer what the model parameters and forcings must have been. E.g. we may have observed record of glacier length fluctuations. This approach has been successfully applied to infer the geothermal heat flux and past mass balance from ice core dating horizons (Grinsted & Dahl-Jensen 2002).

Simple flow models based on the continuity equation with few internal degrees of freedom (e.g. Dansgaard & Johnsen 1969) are generally the preferred method for obtaining age-depth relationships. Ice has a highly non-linear flow law where the effective viscosity is dependent on local stresses, temperature, water content, impurities, grain size, and ice fabric. For these reasons the physics of the ice mass are generally greatly simplified in numerical simulations. A summary of various models of varying complexity can be found in Pattyn (2002). In recent years there has been some progress towards solving the full stokes equations of the flow (Le Meur et al. 2006), even

35

including ice fabric development (Gillet-Chaulet et al. 2004). However, most modelling approaches need information on all the factors influencing the effective viscosity. Usually this is parameterized through flow law parameters such as the so-called ‘flow exponent’, ‘enhancement factor’, and temperature dependent ice rheological parameters (Paterson 1994). These parameters are generally only valid for a particular flow regime and are specific to the area of interest (Paterson 1994).

3.2.1 Antarctic blue ice areas

Blue ice areas (BIAs) are characterized by having exposed ice at the surface due to a negative surface mass balance. Of special interest, are blue ice areas where very localized conditions lead to islands of ablation surrounded by accumulation areas (Bintanja 1999). Many Antarctic BIAs are important stranding surfaces for large numbers of meteorites (Whillans & Cassidy 1983). The age of the surface ice has been determined by radiogenic and meteorite exposure dating to be typically between 10 thousand to 200 thousand years, though some meteorites have terrestrial ages of more than 2 million years (Bintanja 1999, Harvey et al. 1998). In high altitude BIAs melting does not occur and the areas should contain ancient ice at the surface in essentially pristine state. The flow of ice leads under steady state conditions to a continuous progression of isochrones on the surface (see Fig. 7), and we can take a ‘horizontal ice core’ directly from the surface. This valuable source of paleoclimatic data has largely not been utilized to date, largely because of difficulties in dating the ice. Most BIAs are found in the proximity of mountain ranges and nunataks which complicate flow modelling considerably (Bintanja 1999).

Fig. 7. Schematic of a blue ice area. Dashed line show the equilibrium line.

Naruse & Hashimoto (1982) used a simple flow model based on the continuity equation (Nye 1952) to date the Yamato Mountains BIA. Internal flow velocities are calculated from estimates of surface velocities, mass balance, and ice thickness. This type of model has proved to be very useful in dating vertical ice cores. However, Azuma et al. (1985) argue that the assumptions in this model are unrealistic in mountainous regions, and refine the model to take horizontal divergence into account. Surface velocities are inferred from continuity considerations and the fact that the horizontal velocity must be

36

zero where the ice meets the mountains (similarly to the model by Whillans & Cassidy 1983).

4 Objectives of the thesis

Glaciological research in the Arctic Centre focuses on ice cores, Antarctic blue ice areas, sea level, and large scale climate variability. The work in this thesis touches upon all of these subjects and is thus broad in scope. However, my own contribution is clearly defined as the collaborators either are chemists or geophysicists rather than modellers. The main aims of the thesis were:

1. to utilize the ancient ice in Antarctic BIAs for paleoclimatic reconstruction. The main obstacle is obtaining a reliable dating of the ice. In the present work a new ice flow model suitable for BIA dating is presented.

1. to make a new GSL reconstruction which is independent of recent satellite observations and assess the relative importance of mass and density contributions to sea level rise.

2. to extract environmental signals from the Lomonosovfonna ice core and assess whether the environmental signals in the ion records are preserved despite seasonal melt.

3. to examine large scale climatic links between the Tropics and the Arctic using advanced time series techniques.

5 Results

5.1 Dating Antarctic blue ice areas (Paper I)

We present a new simple flow model suitable for dating Antarctic BIAs (Grinsted et al. 2003I ). Contrary to the models by Naruse & Hashimoto (1982) and Azuma et al. (1985) the new model is not based on the theoretical flow law for ice (Glen 1955), but rather an empirical fit to the horizontal velocity profile. The model has a single parameter that allows softening of ice with depth to be taken into account. The model is forced by observed surface velocities, mass balance, and ice thickness (similarly to Naruse & Hashimoto 1982). Any horizontal divergence is accounted for implicitly. The advantage of this model approach is that it is highly constrained by available observations. The model assumes steady state. However, as long as the geometry is constant we can prescribe a history of mass balance and surface velocity changes. This is possible because we calculate particle back trajectories which ensure that the absolute age is known as we follow a particle.

The model is applied to two very different BIAs which we characterize as ‘closed type’ and ‘open type’. The ‘closed’ and ‘open’ refers to whether the flow is forced to terminate at a mountain or whether the flow continues through the BIA into another accumulation area. In the open type Allan Hills, Near Western Ice Field (77°S 159°E 2000 m asl), we can match the age inferred from meteorite finds by lowering both accumulation rates and surface velocities during the last glacial. The model has also been successfully applied on the largest BIA in Antarctica with flow lines as long as 200 km (Moore et al. 2006b).

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Fig. 8. Velocity map in Scharffenbergbotnen. Nunataks are marked in gray, moraines as dark gray, and blue ice as light gray. Stars with arrows show location of stakes and measured velocities. Contours show gridded surface velocities map based on the stake velocities (m/yr). Dotted line show the flow line used in the modeling.

The main BIA in Scharffenbergbotnen (74ºS, 11ºW, 1200 m asl) is a closed type Blue Ice Area as the flow is forced to terminate in the end of the valley (Grinsted et al. 2003I ). In steady state ice flow is only just balanced by net ablation as there is no outflow from the valley. Hence, ice in the innermost part of the valley should be very old, analogous to the situation near the bed in a vertical ice core. The steady state scenario is modeled in Grinsted et al. (2003I ) and show ages greater than 100,000 years. However, the 14C dating of the SBB01 ice core from the innermost part of the main BIA indicate an age of only 10,500±600 years (van der Kemp et al. 2002), which is less than the time needed to travel along the surface from the equilibrium line using present day surface velocities (Error! Reference source not found.). Therefore, SBB can not have been in steady state throughout the Holocene. Here, I explore the possibility that SBB may have been an accumulation area during the Last Glacial Maximum (LGM) and the consequences for the dating. I assume that prior to 20 kyr BP the entire valley had an accumulation rate of 0.2 myr-1. To balance this accumulation I assume a general outflow from the valley, with zero surface velocity in the innermost part of the valley (x=0 km) linearly increasing to 1 myr-1 at valley entrance (x=5 km). The mass balance and the surface velocity are prescribed to linearly change with time to match the present day observations. A comparison of the modelled surface ages with the 14C ages is shown in Fig. 9. Based on this evidence alone it seems plausible that the BIA may have been smaller or even an accumulation area in the past. To examine the past flow regime under different conditions we plan to apply a full 3d thermo-mechanical finite element model of the area (as used by Le Meur et al. 2004).

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Fig. 9. Modeled surface age of Scharffenbergbotnen BIA along the flow line (see Error! Reference source not found.) assuming that the area was an accumulation area in the past. Circles show 14C ages (van Roijen 1996) with errorbars calculated from the distance to the flow line. These errors were estimated by plotting the relative difference between two 14C ages against their distance and fitting a straight line.

5.2 Wavelet coherence (Paper II, Paper VI)

In Grinsted et al. (2004II ) we present the cross wavelet transform (XWT) and wavelet coherence (WTC) and show their application to geophysical time series. A Monte Carlo statistical significance test for WTC against red-noise has been developed. We find that the significance level for the WTC is largely independent of the noise colour, which may imply that a theoretical white noise test can be used. We further show how circular means and circular standard deviations can be used to quantify the relative phase relationship between the series. The methods are illustrated by the relationship between the AO and the BMI and we show that the BMI passively mirrors the atmospheric forcing (i.e. the AO).

To accompany the Grinsted et al. (2004II ) paper, I have developed an open source wavelet coherence toolbox for Matlab, which is freely available on the web (http://www.pol.ac.uk/home/research/waveletcoherence/). We are actively maintaining this software package and also providing support for people using it.

The Quasi Biennial Oscillation (QBO) is a very regular oscillation in the stratosphere with a period of ~28 months. The QBO is conventionally defined as the zonal average of the 30mb zonal wind at the equator. The oscillation propagates to the entire planet with a 3 month lag between pole and equator (Baldwin et al. 2001). AO also has a significant spectral peak at roughly the same period and it has therefore been suggested that QBO drives AO on these wavelengths. In Moore et al. (2006aVI ) we find virtually no wavelet coherence between QBO and AO which casts this hypothesis into doubt.

Distance [km]

Age

[kyr

]

0 1 2 3 4 5 60

2

4

6

8

10

12

14

16

18

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5.3 Lomonosovfonna Melt and Continentality (Paper III)

Chemical records from ice cores that suffer seasonal melting have a reputation of being poor sources of information for paleoclimate reconstructions, (Koerner 1997). However, advances in both analytical and mathematical techniques suggests that certain ice cores preserve reliable records, that can be extracted with confidence, at least on decadal scales (Goto-Azuma et al. 2002, Kekonen et al. 2005a, Pohjola et al. 2002a) – but doubts remain in their interpretation at higher resolution.

In Moore et al. (2005a) we show that the signals in Lomonosovfonna ion record are primarily of environmental origin and that the melt signal caused by melt is mainly explained by a shift in the mean concentrations. We compare ion composition between bubbly ice layers and clear ice (clear ice is indicative of refrozen melt water) using principal component analysis. Only the first principal component (PC) shows significant differences between the two groups. When we compare the ion composition between different climatic periods we find significant differences in the first 4 PCs.

Many studies have found that particular ions are preferentially washed out of the ice during seasonal melt (e.g. Davies et al. 1982, Iizuka et al. 2002). In Grinsted et al. (2006aIII ) we build a model for the chemical fractionation during melt. Based on this model we obtain a proxy for melt intensity based on the logarithmic ratio between two ions with the same principal source but different elution rates. We use two ion pairs (Na+/Mg2+ and Cl-/K+) to obtain two independent melt proxies and show that their records agree qualitatively. In addition to summer temperature, the melt proxies also appears to reflect sea ice extent, likely as a result of sodium chloride fractionation in the oceanic sea ice margin source area that is dependent on winter temperatures.

The amplitude of the annual signal in δ18O (A) is an obvious candidate for a continentality proxy, since δ18O on Lomonosovfonna is a proxy for surface air temperature (Isaksson et al. 2001, van de Wal et al. 2002) and the annual cycle is well preserved (Pohjola et al. 2002b). However, melt and firn diffusion processes alter the deposited signal which complicates the interpretation of the amplitudes. With modelling and through correlations with instrumental temperature range we show that the amplitude record can be used as a continentality proxy provided that it is smoothed sufficiently (Grinsted et al. 2006aIII ).

The melt proxies, the δ18O, and the continentality proxy tell a consistent story of a colder and more continental climate with larger sea ice extent during the Little Ice Age in Svalbard (Grinsted et al. 2006aIII ). Perhaps the most surprising finding is that continentality decreased around 1870, 20–30 years before the rise in temperatures indicated by the δ18O profile. The melt record indicates that summers were warmer in the Medieval Warm Period than during the 1990s – but maybe similar to post-2000 summers.

5.4 Sea level (Paper IV & VIII)

Producing a GSL curve with valid confidence intervals from the observational data base is not a trivial job. We have developed a new ‘virtual station’ method to overcome geographical bias, so that stations close to each other are weighted much less than

42

isolated ones, and which can quantify the uncertainties due to the representativeness of the stations used (Jevrejeva et al. 2006, Grinsted et al. 2006bIV ).

As there is no common reference level for the tide gauge data, we examine the rate of change in sea level rather than sea level itself. Our approach for creating a GSL curve is therefore to integrate the rate of change in GSL (dGSL), which we obtain by making a geographically weighted average (see below) of the sea level rates from individual tide gauge stations. Prior to analysis modelled GIA corrections were applied to the tide gauge records.

We assign each station to one of 13 regions. We then recursively collapse the two closest stations within a region (by averaging their records) into a new virtual station half-way between them until only one station remains. This last remaining virtual station represents the average for the entire region. This ensures that isolated tide gauge records are given more weight. Whenever a virtual station is created the uncertainty due representativity can be calculated by looking at the deviation from the mean of the source stations over the period of common overlap (Jevrejeva et al. 2006). Finally, the GSL is calculated as the integrated average of the regional sea level rates excluding the Baltic region (Fig. 10).

Fig. 10. Global sea level relative to the 20th century mean, calculated using the virtual station method (thin solid line) and its SSA non-linear trend with 10 year lag (thick line). Grey band shows the 95% confidence interval of the trend (a jack-knife estimate).

It has previously been noted that there are drops in global sea level (GSL) following some major volcanic eruptions (Church et al. 2004, Church & White 2006). However, observational evidence has not been convincing as there is substantial variability in the global sea level record on periods similar to those at which we expect volcanoes to have an impact (Jevrejeva et al. 2005). In Grinsted et al. (2006bIV ) we quantify the impact of volcanic eruptions by averaging the virtual station GSL around 5 major volcanic eruptions. Surprisingly, we find that the initial response to a volcanic eruption is a significant rise in sea level of 9 mm in the first year after the eruption. This is followed by a drop of 7 mm in the period 2-3 years after the eruption relative to pre-eruption sea level.

GS

L (m

m)

Year (A.D.)1880 1900 1920 1940 1960 1980 2000

−300

−200

−100

0

100

200

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Neither the drop nor especially the rise in GSL can be explained by models of lower oceanic heat content (Hansen et al. 2002, Church et al. 2005). We suggest that the largest volcanic impact is a transient disturbance of the water cycle, comparable to the impact of a large El Niño-La Niña cycle, amounting to about 5% of global land precipitation.

5.5 Noise reduction in Ground Penetrating Radar (Paper V)

Ground Penetrating Radar (GPR) is a commonly used tool in glaciology. It is frequently used for finding the total ice thickness, the depth of the water table, and for following internal isochrones (e.g. Sinisalo et al. 2003). In glaciological research, there are frequently prohibitive logistical difficulties in returning to a particular glacier to collect additional data that may aid interpretation. Therefore it is worth going to more trouble than would be the case in a commercial survey, to extract the maximum glaciological information from existing radar data.

In Moore & Grinsted (2006V ) we apply SSA for noise reduction. We combine this noise reduction with an envelope detection scheme followed by horizontal stacking of the radar traces. Our methods show a dramatic improvement in visual quality of a low signal to noise ratio GPR data set compared with traditional methods.

5.6 ENSO-Arctic links (Paper VII)

Jevrejeva et al. (2003) found what appeared to be a common ~14 year signal in both the AO and the SOI using singular spectrum analysis. In Jevrejeva et al. (2004VI ) we isolate the 14 year signal from 4 different Arctic series and 4 different ENSO series, and they all appear to be phase locked. We find that SOI is leading the Arctic series by roughly 2 years. This result is confirmed by wavelet coherence.

We develop a new method to track how the 14 year signal isolated from SOI propagates around the world in gridded sea surface temperatures. The phase coherence and the average relative phase between SST in a grid cell and the isolated signal are calculated. In this way we plot regions of significant coherence and the relative lag within these regions. The observed pattern in coherence and phase lags can be explained by equatorial coupled wave propagation in the Pacific Ocean followed by Kelvin boundary wave propagation along the western margins of the Americas and by poleward-propagation of atmospheric angular momentum (Jevrejeva et al. 2004VI ). The method of mapping the phase coherence and mean relative phase can be considered a ‘phase-aware teleconnection’ pattern.

5.7 Sunspots and climate (Paper VI)

There is considerable dispute as to the strength of the sun-climate link (Tsiropoula 2003, Laut 2003). Many proxy climatic time series exhibit significant power in the 11-14 year

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band, which several authors have been tempted to ascribe to solar sunspot cycles (e.g. Rigozo et al. 2002).

In Moore et al. (2006a) we examine the plausibility of the argument that solar cycles are significant factors in climate on multi-year and decadal timescales. The lagged phase coherence between sunspot numbers and major climate indices (SOI, Niño3.4, and AO) at the 11 year period shows no signs of sunspots being a driving mechanism. We therefore conclude that there is no simple causative relationship between sunspot numbers (and hence solar insolation) and multi-year to decadal signals in the large circulation patterns that define in large planet’s climate. In Fig. 11 the phase-aware teleconnection pattern between sunspot numbers and global SST is shown. The lack of coherence and random phase orientations confirm the results of Moore et al. (2006aVI ).

Fig. 11. Lack of pattern in phase-aware teleconnection between SST (Smith & Reynolds 2004) and sunspot numbers over the 20th century using a Paul wavelet with 11 year centre periodicity. Regions of significant coherence are shown with a thick black contour and light grey shading. Arrows indicate the average phase relationship (pointing right: in-phase; left: anti-phase; down: sunspots lagging by 90°). For clarity phase arrows are not plotted where the phase coherence is below 0.2.

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Original papers

The following papers comprise the work in this thesis:

I Grinsted A, Moore J, Spikes VB & Sinisalo A (2003) Dating Antarctic blue ice areas using a novel ice flow model. Geophys Res Lett 30(19). doi:10.1029/2003GL017957.

II Grinsted A, Moore JC & Jevrejeva S (2004) Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlin Process Geophys 11: 561-566. (http://www.pol.ac.uk/home/research/waveletcoherence/).

III Grinsted A, Moore JC, Pohjola V, Martma T & Isaksson E (2006a) Svalbard summer melting, continentality, and sea ice extent from the Lomonosovfonna ice core. J Geophys Res 111: D07110. doi:10.1029/2005JD006494.

IV Grinsted A, Moore JC & Jevrejeva S (2006b) Observational evidence for volcanic impact on sea level and the global water cycle. (Manuscript).

V Moore & Grinsted (2006) Singular spectrum analysis and envelope detection: methods of enhancing the utility of ground penetrating radar data. J Glaciol 52(176): 159-163(5).

VI Moore J, Grinsted A & Jevrejeva S (2006a) Is there evidence for sunspot forcing of climate at multi-year and decadal periods? Geophys Res Lett 33: L17705. doi:10.1029/2006GL026501.

VII Jevrejeva S, Moore JC & Grinsted A (2004) Oceanic and atmospheric transport of multi-year ENSO signatures to the polar regions. Geophys Res Lett 31: L24210. doi:10.1029/2004GL020871.

VIII Jevrejeva S, Grinsted A, Moore JC & Holgate S (2006c) Nonlinear trends and multiyear cycles in sea level records. J Geophys Res 111: C09012. doi:10.1029/2005JC003229.

Reprinted with permission of the American Geophysical Union (I, III, VI, VII, VIII), and the International Glaciological Society (V).

Original publications are not included in the electronic version of the dissertation.

I

Dating Antarctic blue ice areas using a novel ice flow model

Aslak Grinsted,1,2 John Moore,1 Vandy Blue Spikes,3 and Anna Sinisalo1,2

Received 13 June 2003; accepted 8 September 2003; published 11 October 2003.

[1] We present a new type of flow model suitable forAntarctic blue ice areas, with application to dating ice forpaleoclimate purposes. The volume conserving model usesfield data for surface velocities, mass balance and icethickness along a flow line, with parameterized variation ofice rheology with depth to produce particle trajectories andisochrones. The model is tested on the contrasting AllanHills Near Western Ice Field and the Scharffenbergbotnenblue ice fields in Antarctica by comparing predicted ageswith ages inferred from meteorites and 14C data. During theglacial periods, ice surface velocities at the Allan Hills musthave been 25% less, and accumulation rates 50% less thanpresent day values in order to match meteorite ages. Incontrast, Scharffenbergbotnen velocities have probably beenfairly constant over time due to the atypical valley where itresides. INDEX TERMS: 1827 Hydrology: Glaciology (1863);

1863 Hydrology: Snow and ice (1827); 3210 Mathematical

Geophysics: Modeling; 9310 Information Related to Geographic

Region: Antarctica. Citation: Grinsted, A., J. Moore, V. B.

Spikes, and A. Sinisalo, Dating Antarctic blue ice areas using a

novel ice flow model, Geophys. Res. Lett., 30(19), 2005,

doi:10.1029/2003GL017957, 2003.

1. Introduction

[2] Blue ice areas (BIAs) are characterized by havingexposed ice at the surface due to a negative surface massbalance. Of special interest, are blue ice areas where verylocalized conditions lead to islands of ablation surroundedby accumulation areas. Many Antarctic BIAs are importantstranding surfaces for large numbers of meteorites [Whillansand Cassidy, 1983]. The age of the surface ice has beendetermined by radiogenic and meteorite exposure dating tobe typically between 10 thousand to 200 thousand years,though some meteorites have terrestrial ages of more than2 million years [Bintanja, 1999; Harvey et al., 1998]. Inhigh altitude BIAs melting does not occur and the areasshould contain ancient ice at the surface in essentiallypristine state. This valuable source of paleoclimatic datahas not been utilized to date, largely because of difficultiesin dating the ice. In this paper we present a modelingapproach that requires limited, and usually available inputdata that can be used in BIAs that have rather complicatedflow regimes - as in fact do many of them.[3] For the purposes of modeling, a useful classification

is in terms of whether the ice flow is forced to terminatebecause of outcropping nunataks (closed type), or whether

the flow is free (open type). The closed type must have veryold surface ice closest to the nunatak where horizontal icevelocity goes to zero, while the open type will haverelatively young surface ice as velocities never go to zero.[4] First we introduce the flow line model, and then use it

to date surface ice on a typical open BIA in West Antarcticaand a typical closed BIA in East Antarctica. Modeled agesare compared with radiogenic and meteorite exposure ages,and the effects of changing climate on the ice flow regimesexamined.

2. Model

[5] A full three dimensional model is always the bestapproach for ice sheet modeling, however, the necessaryinput field data are generally unavailable, or limited intemporal coverage. It is much more practical to use a flowline approach where sparse field data can be most easilyincorporated into the model. Traditional flow line modelshave either been based on the shallow ice approximation[e.g., Paterson, 1994] or the full stress equilibrium equa-tions [e.g., Pattyn, 2002]. The shallow ice type models haveproblems in areas such as Scharffenbergbotnen where theflow is not always down slope, bedrock relief is large or theflow is constrained by valley sides. Models based on the fullstress equilibrium equations are very good for predictinghow the ice reacts to mass balance perturbations. However,these models are not fully constrained to the observed data.Attempts to model flow in specific BIAs have also usedapproaches based on parameterized ice thickness and diver-gence terms [Whillans and Cassidy, 1983; Azuma et al.,1985; van Roijen, 1996]. In this paper we introduce a simplemodel based on volume conservation and force it with theobserved surface velocity, ice thickness and mass balance.[6] We choose x to be the distance along the flow line, z

as the water equivalent height above the bed and y isperpendicular to the flow. The model is formulated usinga vertically scaled coordinate z = z/H, where H is the waterequivalent ice thickness. The velocity components of x, y, zand z are U, V, W and w. The horizontal velocity can bewritten as

U zð Þ ¼ f zð ÞUs and V zð Þ ¼ 0;

where subscript ‘s’ denote the value at the surface. In thispaper we assume that ice is frozen at the bed. Empiricallywe find a reliable approximation is

f zð Þ ¼ tanh kzð Þtanh kð Þ :

The value of f, and hence k should reflect the softening ofthe ice with depth. The rheological behavior of ice dependson its crystal fabric, impurity content and temperature. In

GEOPHYSICAL RESEARCH LETTERS, VOL. 30, NO. 19, 2005, doi:10.1029/2003GL017957, 2003

1Arctic Centre, University of Lapland, Rovaniemi, Finland.2Department of Geophysics, University of Oulu, Oulu, Finland.3Department of Earth Sciences and the Climate Change Institute,

University of Maine, Orono, Maine, USA.

Copyright 2003 by the American Geophysical Union.0094-8276/03/2003GL017957$05.00

CRY 1 -- 1

Antarctica changes due to impurities and crystal fabricsseem much less important than those caused by thetemperature profile within the ice sheet [Paterson, 1994].The steady state temperature profile in an ice sheet is linearat the equilibrium line, and as we are dealing with slow flowon each side of the equilibrium line a linear profile isreasonable for the whole flow line (Figure 1). Therefore, wewrite

@U

@x¼ f zð Þ @Us

@xand

@V

@y¼ f zð Þ @Vs

@y:

From volume conservation of an ice column we get

@Us

@xþ @Vs

@y¼ 1

fm� @Um

@xþ @Vm

@y

� �¼ b

fmH;

where b is the mass balance and subscript ‘m’ denote amean over the entire ice column. The continuity equationassuming incompressibility is written

@U

@xþ @V

@yþ @W

@z¼ 0 ) @W

@z¼ � @Us

@xþ @Vs

@y

� �� f zð Þ:

Hence

w zð Þ ¼ � @Us

@xþ @Vs

@y

� ��Zz

0

f z0ð Þdz0

¼@Us

@xþ @Vs

@y

�k tanh kð Þ log1þ e�2kz

2

� �þ kz

� �:

As input data b, H and Us are prescribed as functions of x. ALagrangian integration scheme is then used to calculateparticle back trajectories and thus the dating.[7] This new volume conservation approach with a

simple yet realistic velocity profile makes use of all thedata typically available. Since the model is so simple, thereis basically no room for tuning. We now apply the model totwo different BIAs.

3. Allan Hills, Near Western Ice Field (NWIF)

[8] The Allan Hills NWIF (77�S 159�E 2000 m asl) flowline starts on a typical snow accumulation area, passesthrough the BIA and then continues through downstreamsnowfields. The flow line, surface velocities, surface ele-vations and mass balance data (Figure 2) were found from asurvey network which was established in 1997 and resur-veyed �2 years later. Ice thickness comes from Desisle andSievers [1991]. Flow is essentially parallel, with littledivergence.[9] Figure 2 shows the particle paths and isochrones

along the flow line. It is remarkable that all the surfaceice in the BIA was originally deposited within a few km ofthe equilibrium line. None of the surface ice has been within150 m of the bedrock. Sensitivity tests with doubled andhalved accumulation rates show little difference in surfaceages and origin of the ice, whereas changing the surfacevelocities have a much larger impact.[10] The modeled maximum age of surface ice is approx-

imately 60 thousand years, using present day mass balanceand velocities (Figure 3). For comparison, the maximumterrestrial age of meteorites found on the NWIF is 180 ±20 thousand years (Kunihiko Nishiizumi, unpublished da-ta). This suggests that accumulation rates and surfacevelocities have been significantly lower in the past. Deepice cores and boreholes suggest that surface temperatures

Figure 1. Normalized plot of height above bed against icevelocity. Blue dashed curves are profiles found using themodel f, red curves are found using the shallow iceapproximation using flow parameters based on the compila-tion of Paterson [1994]. The model f can approximate awide range of velocity profiles through an appropriatechoice of k. The linear temperature profile is for a surfacetemperature of �30�C and bed temperature of �10�C. Wechose k = 5 to model both BIAs in the paper.

Figure 2. Top: Present day mass balance (dashed blue)and surface velocity (red) along the NWIF flow line.Bottom: Particle paths (dotted lines) and isochrones(colored contours) for the NWIF flow line with accumula-tion rates 50% and surface velocity 25% of present dayvalues from 11500–115000 and before 125000 years BP.

CRY 1 - 2 GRINSTED ET AL.: DATING ANTARCTIC BLUE ICE AREAS

were about 10–15�C cooler than present, and accumulationrates were about 50% lower during glacial times in Ant-arctica [Paterson, 1994]. Equilibration times for the icesheet to respond to changes in climate are only in the orderof a few thousands of years [Naslund et al., 2000], and areduction in temperature of the ice sheet by 10�C increasesits stiffness by about 3.5 times [Paterson, 1994]. Slightchanges in ice thickness could result in different surfaceslopes and thus velocities. As shown in Figure 3 themaximum age of the surface ice changes from about60 thousand years, using the model with present dayvelocities and mass balance, to 190 thousand years, byreducing accumulation rates by half and ice velocities to25% during the glacial periods. Ablation rates are assumedto be constant, which is justified by the similarity betweenthe ablation rates at both BIAs modeled here, despite largedifferences in surface temperature.

4. Heimefrontfjella, Scharffenbergbotnen (SBB)

[11] SBB (74�S, 11�W, 1200 m asl) is the best-studiedblue ice area in Antarctica from a glaciological point ofview. The flow line (Figure 4) was picked from griddedsurface velocity data [Sinisalo et al., 2003; van Roijen,1996]. Mass balance data (Figure 5) comes from stakeobservations [Sinisalo et al., 2003], 14C measurements[van Roijen, 1996] and radar internal reflections, surfacetopography from Sinisalo et al. [2003] and ice thicknessfrom [Herzfeld and Holmlund, 1990].[12] In contrast to earlier modeling of flow in SBB [van

Roijen, 1996], it is clear that the ice does not originate froman ice divide at the entrance to the valley (Figure 4), butsome comes with the main ice flux from the plateau. Themain influx enters through the northern entrance of thevalley, with only minor inflows via the shallow and narrowsouthwestern portal and the eastern end. It is obvious thatthe valley is most atypical in terms of its low accumulationrate (on the snow covered areas), and very low ice veloc-ities. Wind turbulence inhibits snow accumulation, and

since it is a closed BIA, ice flow only just balances netablation.[13] The flow line passes over a bedrock high that is an

extension of the exposed nunataks along the northeasternside of the valley. Much of the flow from the higherelevation accumulation area does not flow into the valley,but passes across the valley entrance and then to thenorthwest. The ice that enters the valley diverges as theflow line turns south, and slows considerably. The flow ofice into the valley is regulated by the extent of thedivergence, which is heavily influenced by the ice sheetmass balance. During periods of ice sheet thickening (asmodeled after the end of the glacial period [Naslund et al.,

Figure 3. Comparison of surface ages along the NWIFflow line. The dashed blue curve assumes present dayaccumulation rates and surface velocities (from Figure 2),solid red uses reduced values in the glacial period.

Figure 4. SBB gridded velocity data based on stakemeasurements (X). The flow line (dotted), the visible extentof BIAs (blue), moraines (grey) and nunataks (brown) aremarked.

Figure 5. Top: Mass balance (dashed blue) and surfacevelocity (red) for the SBB flow line. Bottom: Particle paths(dotted lines) and isochrones (colored contours) for the SBBflow line.

GRINSTED ET AL.: DATING ANTARCTIC BLUE ICE AREAS CRY 1 - 3

2000; van Roijen, 1996]), more ice flows into the valley,thereby raising the surface, and eventually inhibiting iceinflow. During periods of thinning, less ice flows into thevalley until ablation of the BIA lowers the surface, therebycompensating for reduced influx. In this way the icevelocity inside the valley is not strongly determined bythe ice rheology variations caused by changing climate,though the ice flow and accumulation rate outside the valleymust be instep with widespread changes. Therefore, in ourmodel, the mass balance and surface velocity fields are leftunchanged over time.[14] Figure 5 shows the particle paths and isochrones

along the flow line. Ice from the first 500 m of the flow line(>50 thousand years old) originates from more than 10 kmaway. The northern small BIA (Figure 4) contains relativelyyoung ice as it is an open-type BIA, and accumulation ratesand velocities outside the valley [Naslund et al., 2000] aremuch higher than within it (Figure 5). The small accumu-lation area inside the valley supplies ice that is up to20 thousand years old to the main BIA. The particle pathsof the oldest ice nearly reach the bedrock. However, theynever come closer than 11 m. It is unclear whether this isclose enough to the bed for the ice chronology to have beendisturbed. Results from deep ice cores in Greenland [Tayloret al., 1993] suggest that the bottom-most layers are mixedon scales of at least 10 cm.[15] The most striking result is the discrepancy between

the model ages and the 14C ages (Figure 6). Old model agesare an inevitable result of velocities going to zero at the endof the valley. The only way to produce ages comparable tothe 14C ages is to make the whole valley an accumulationarea in the recent past (�8000 years BP). However, thisleads to an essentially constant surface age for the BIA,which is in conflict with observations of dipping radarlayers and visible surface bands. This discrepancy may be

because of unaccounted 14C processes such as in situproduction by muons [van der Kemp et al., 2002] or CO2

release from organic compounds [Colussi and Hoffmann,2003]. However, a young age could explain the lack ofmeteorite finds on this BIA.

5. Conclusion

[16] The flow model has the advantage of being bothflexible and simple, allowing realistic particle trajectories tobe calculated at arbitrary resolution. The model makes useof input velocity, thickness and mass balance data in a self-consistent way. One significant advantage of using mea-sured surface velocities is that the details of the ice rheologydo not need to be considered, beyond estimation of the kparameter. In practice, it is never possible to know the realvariations in rheology along a flow line, so this approach isas good as attempting to model rheology directly.[17] The model does assume volume conservation - i.e.

that the ice thickness does not change with time - this is easyto modify if there is a good reason to suppose that icethickness has changed. However, consideration of at leastthe SBB example of a closed type BIA suggests thatthickness variations over time are very small. The modeldoes accommodate temporally variable surface velocity andmass balance.[18] Results for NWIF BIA show that open type areas can

contain ice as old as 200 thousand years, which is actuallycomparable with the oldest ice in SBB. The oldest ice fromSBB may be disturbed due to flowing within 11 m of thebedrock, so a better climate record may be available fromNWIF and open-type ice fields in general. However, theflow in open-type ice fields is more sensitive to climatechange, thus making them harder to date. In contrast tomany assumptions regarding accumulation areas for BIAs,the two examples presented here have small local catch-ments. Short flowlines makes paleoclimate interpretationsof records from the BIAs much easier.

[19] Acknowledgments. Funding was provided by Thule Instituteand Finnish Academy through FINNARP. Fieldwork at AH was performedby Ian Whillans and VBS and funded by the United States National ScienceFoundation.

ReferencesAzuma, M., N. Nakawo, A. Higashi, and F. Nishio, Flow pattern nearMassif A in the Yamato bare ice field estimated from the structuresand the mechanical properties of a shallow ice core, Mem. NIPR, Spec.Iss., 39, 173–183, 1985.

Bintanja, R., On the glaciological, meteorological, and climatological sig-nificance of Antarctic blue ice areas, Rev. Geophys., 37, 337–359, 1999.

Colussi, A. J., and M. R. Hoffmann, In situ photolysis of deep ice corecontaminants by Cerenkov radiation of cosmic origin, Geophys. Res.Lett., 30(4), 1195, doi:10.1029/2002GL016112, 2003.

Desisle, G., and J. Sievers, Sub-ice topography and meteorite finds near theAllan Hills and the Near Western Ice Field, Victoria Land, Antarctica,J. Geophys. Res., 96, 15,577–15,587, 1991.

Harvey, R. P., N. W. Dunbar, W. C. Macintosh, R. P. Esser, K. Nishiizumi,S. Taylor, and M. W. Caffee, Meteoric event recorded in Antarctic ice,Geology, 26(7), 607–610, 1998.

Herzfeld, U. C., and P. Holmlund, Geostatistics in glaciology: Implicationsof a study of SBB, Dronning Maud Land, East Antarctic, Ann. Glaciol.,14, 107–110, 1990.

Naslund, J. O., J. L. Fastook, and P. Holmlund, Numerical modelling of theice sheet in western Dronning Maud Land, East Antarctica: Impacts ofpresent, past and future climates, J. Glaciol., 46, 54–66, 2000.

Paterson, W. S. B., The Physics of Glaciers, 3rd edn, Butterworth-Heinemann, Woburn, Mass., 1994.

Figure 6. Comparison of surface ages along the SBB flowline (blue line) with 14C ages (converted to calendar yearswith 1s error bars, marked in red). The rightmost section ofthe blue line represent the surface ice in the small BIA,middle section ice in the main BIA originating from theaccumulation area inside the valley and the left section iceoriginating from outside the valley.

CRY 1 - 4 GRINSTED ET AL.: DATING ANTARCTIC BLUE ICE AREAS

Pattyn, F., Transient glacier response with a higher-order numerical ice-flowmodel, J. Glaciol., 48, 467–477, 2002.

Sinisalo, A., J. C. Moore, R. V. D. Wal, R. Bintanja, and S. Jonsson, A14-year mass balance record of a blue ice are in Antarctica, Ann.Glaciol., 37, in press, 2003.

Taylor, K. C., C. U. Hammer, R. B. Alley, H. B. Clausen, D. Dahl-Jensen,A. J. Gow, N. S. Gundestrup, J. Kipfstuhl, J. C. Moore, and E. D.Waddington, Ice flow and the climate record at Summit, Greenland,Nature, 366, 549–552, 1993.

van der Kemp, W., C. Alderliesten, K. van der Borg, A. F. M. de Jong,R. A. N. Lamers, J. Oerlemans, M. Thomassen, and R. S. W. van der Wal,In situ produced 14C by cosmic ray muons in ablating Antarctic ice,Tellus, 45B, 186–192, 2002.

van Roijen, J. J., Determination of ages and specific mass balances from14C measurements on Antarctic surface ice, Ph.D. thesis, Faculty ofPhysics and Astronomy, Utrecht Univ., Utrecht, 1996.

Whillans, I. M., and W. A. Cassidy, Catch a falling star; meteorites and oldice, Science, 222, 55–57, 1983.

�����������������������A. Grinsted, J. Moore, and A. Sinisalo, Arctic Centre, University of

Lapland, P.O. Box 122, FIN-96101 Rovaniemi, Finland. (ag@glaciology.net)V. B. Spikes, Department of Earth Sciences and the Climate Change

Institute, University of Maine, Orono, ME 04469-5790, USA.

GRINSTED ET AL.: DATING ANTARCTIC BLUE ICE AREAS CRY 1 - 5

II

Nonlinear Processes in Geophysics (2004) 11: 561–566SRef-ID: 1607-7946/npg/2004-11-561 Nonlinear Processes

in Geophysics© European Geosciences Union 2004

Application of the cross wavelet transform and wavelet coherence togeophysical time series

A. Grinsted1, J. C. Moore1, and S. Jevrejeva2

1Arctic Centre, University of Lapland, Rovaniemi, Finland2Proudman Oceanographic Laboratory, Birkenhead, UK

Received: 11 June 2004 – Revised: 19 October 2004 – Accepted: 16 November 2004 – Published: 18 November 2004

Part of Special Issue “Nonlinear analysis of multivariate geoscientific data – advanced methods, theory and application”

Abstract. Many scientists have made use of the waveletmethod in analyzing time series, often using popular freesoftware. However, at present there are no similar easy touse wavelet packages for analyzing two time series together.We discuss the cross wavelet transform and wavelet coher-ence for examining relationships in time frequency space be-tween two time series. We demonstrate how phase anglestatistics can be used to gain confidence in causal relation-ships and test mechanistic models of physical relationshipsbetween the time series. As an example of typical data wheresuch analyses have proven useful, we apply the methods tothe Arctic Oscillation index and the Baltic maximum seaice extent record. Monte Carlo methods are used to assessthe statistical significance against red noise backgrounds. Asoftware package has been developed that allows users toperform the cross wavelet transform and wavelet coherence(http://www.pol.ac.uk/home/research/waveletcoherence/).

1 Introduction

Geophysical time series are often generated by complexsystems of which we know little. Predictable behavior insuch systems, such as trends and periodicities, is thereforeof great interest. Most traditional mathematical methodsthat examine periodicities in the frequency domain, such asFourier analysis, have implicitly assumed that the underly-ing processes are stationary in time. However, wavelet trans-forms expand time series into time frequency space and cantherefore find localized intermittent periodicities. There aretwo classes of wavelet transforms; the Continuous WaveletTransform (CWT) and its discrete counterpart (DWT). TheDWT is a compact representation of the data and is par-ticularly useful for noise reduction and data compressionwhereas the CWT is better for feature extraction purposes.

Correspondence to: A. Grinsted(ag@glaciology.net)

As we are interested in extracting low s/n ratio signals intime series we discuss only CWT in this paper. While CWTis a common tool for analyzing localized intermittent os-cillations in a time series, it is very often desirable to ex-amine two time series together that may be expected to belinked in some way. In particular, to examine whether re-gions in time frequency space with large common powerhave a consistent phase relationship and therefore are sug-gestive of causality between the time series. Many geophys-ical time series are not Normally distributed and we suggestmethods of applying the CWT to such time series. From twoCWTs we construct the Cross Wavelet Transform (XWT)which will expose their common power and relative phasein time-frequency space. We will further define a measureof Wavelet Coherence (WTC) between two CWT, which canfind significant coherence even though the common power islow, and show how confidence levels against red noise back-grounds are calculated.

We will present the basic CWT theory before we moveon to XWT and WTC. New developments such as quanti-fying the phase relationship and calculating the WTC sig-nificance level will be treated more fully. When using themethods on time series it is important to have solid mecha-nistic foundations on which to base any relationships found,and we caution against using the methods in a “scatter-gun”approach (particularly if the time series probability densityfunctions are modified). To illustrate how the various meth-ods are used we apply them to two data sets from meteo-rology and glaciology. Finally, we will provide links to aMatLab software package.

2 Data

An example of two physical effects that we expect to belinked from consideration of the climate system are the meanwinter state of the arctic atmosphere and winter severity re-flected by ice conditions.

562 A. Grinsted et al.: Cross wavelet and wavelet coherence

Fig. 1. The standardized time series of winter (DJF) AO (bottom)and its continuous wavelet power spectrum (top). The thick blackcontour designates the 5% significance level against red noise andthe cone of influence (COI) where edge effects might distort thepicture is shown as a lighter shade. The standardized AO series hasan AR1 coefficient of 0.02.

The Arctic Oscillation (AO) is a key aspect of climate vari-ability in the Northern Hemisphere. The AO is defined asthe leading empirical orthogonal function (EOF) of North-ern Hemisphere sea level pressure anomalies pole ward of20◦N (Thompson and Wallace, 1998), and characterized byan exchange of atmospheric mass between the Arctic andmiddle latitudes. The Baltic Sea is a transition zone betweenthe North Atlantic region and the continental area of Eura-sia, leading to large inter-annual variability of ice conditions.The Baltic Sea is partly covered by ice every winter season,the maximum annual ice extent varies between 10–100% ofthe sea area, the length of ice season is 4–7 months, and themaximum annual thickness of ice is 50–120 cm (Jevrejeva,2001; Seina et al., 2001). Recently published results demon-strate that large-scale atmospheric circulation patterns in theArctic and North Atlantic described by the AO or by thesomewhat similar North Atlantic Oscillation teleconnectionssignificantly control ice conditions in the Baltic Sea (Loeweand Koslowski, 1998; Omstedt and Chen, 2001; Jevrejevaand Moore, 2001; Jevrejeva, 2002).

In this paper, we examine the connection between win-ter AO and Baltic Sea ice extent and especially explore thephase relationships between the series in the light of the ex-pected causality links. Ice conditions are represented by thetime series of maximum annual ice extent in the Baltic Sea(BMI) for the period 1720–2000 (Seina et al., 2001). Weuse the winter AO index (December–February 1851–1997)of Thompson and Wallace (1998).

Many statistical tests assume that the probability densityfunction (pdf) is close to Normal. Our experience withCWTs of geophysical time series shows that series far fromnormally distributed produces rather unreliable and less sig-nificant results. Occasionally it can therefore be a good idea

Fig. 2. The standardized BMI percentile time series (bottom) and itscontinuous wavelet power (top). The thick contour designates the5% significance level against red noise and the cone of influence(COI) where edge effects might distort the picture is shown as alighter shade. The standardized BMI percentile series has an AR1coefficient of 0.08.

to transform the pdf of the time series. However, we cau-tion against rashly changing the pdf. The BMI index isbi-modally distributed with maximum probabilities around70 000 km2 and 420 000 km2. A simple periodic oscillationbetween these two modes will almost have the shape of asquare wave. The power of a square wave leaks into fre-quency bands outside the fundamental period. We thereforetransform the BMI index into a record of percentiles (in termsof its cumulative distribution function) and thus forcing thepdf to be rectangular. This has the effect of narrowing thebandwidth of intermittent oscillations. We standardize (zeromean, unit standard deviation) both time series and will re-fer to the standardized versions as simply AO and BMI. Thetime series of the AO and BMI time series are shown inFigs. 1 and 2.

3 Methods

3.1 The Continuous Wavelet Transform (CWT)

A wavelet is a function with zero mean and that is localizedin both frequency and time. We can characterize a waveletby how localized it is in time (�t) and frequency (�ω or thebandwidth). The classical version of the Heisenberg uncer-tainty principle tells us that there is always a tradeoff betweenlocalization in time and frequency. Without properly defining�t and �ω, we will note that there is a limit to how small theuncertainty product �t·�ω can be. One particular wavelet,the Morlet, is defined as

ψ0(η) = π−1/4eiω0ηe− 1

2η2. (1)

where ω0 is dimensionless frequency and η is dimensionlesstime. When using wavelets for feature extraction purposes

A. Grinsted et al.: Cross wavelet and wavelet coherence 563

the Morlet wavelet (with ω0=6) is a good choice, since itprovides a good balance between time and frequency local-ization. We therefore restrict our further treatment to thiswavelet, although the methods we present are generally ap-plicable (see, e.g. Foufoula-Georgiou, 1995).

The idea behind the CWT is to apply the wavelet as a band-pass filter to the time series. The wavelet is stretched in timeby varying its scale (s), so that η=s·t, and normalizing it tohave unit energy. For the Morlet wavelet (with ω0=6) theFourier period (λwt ) is almost equal to the scale (λwt=1.03 s).The CWT of a time series (xn, n=1,. . . ,N) with uniform timesteps δt, is defined as the convolution of xn with the scaledand normalized wavelet. We write

WXn (s) =

√δts

N∑n′=1

xn′ψ0[(

n′ − n)

δts

]. (2)

In practice it is faster to implement the convolution in Fourierspace (see details in Torrence and Compo, 1998). We definethe wavelet power as |WX

n (s)|2. The complex argument ofWX

n (s) can be interpreted as the local phase.The CWT has edge artifacts because the wavelet is not

completely localized in time. It is therefore useful to intro-duce a Cone of Influence (COI) in which edge effects cannot be ignored. Here we take the COI as the area in whichthe wavelet power caused by a discontinuity at the edge hasdropped to e−2 of the value at the edge.

The statistical significance of wavelet power can be as-sessed relative to the null hypotheses that the signal is gener-ated by a stationary process with a given background powerspectrum (Pk). Many geophysical time series have distinc-tive red noise characteristics that can be modeled very wellby a first order autoregressive (AR1) process. The Fourierpower spectrum of an AR1 process with lag-1 autocorrela-tion α (estimated from the observed time series e.g. Allenand Smith, 1996) is given by

Pk = 1 − α2∣∣1 − αe−2iπk∣∣2

, (3)

where k is the Fourier frequency index.The wavelet transform can be thought of as a consecutive

series of band-pass filters applied to the time series where thewavelet scale is linearly related to the characteristic periodof the filter (λwt ). Hence, for a stationary process with thepower spectrum Pk the variance at a given wavelet scale, byinvocation of the Fourier convolution theorem, is simply thevariance in the corresponding band of Pk . If Pk is sufficientlysmooth then we can approximate the variance at a given scalesimply with Pk using the conversion k−1=λwt . Torrence andCompo (1998) use Monte Carlo methods to show that thisapproximation is very good for the AR1 spectrum. They thenshow that the probability that the wavelet power, of a processwith a given power spectrum (Pk), being greater than p is

D

(∣∣WXn (s)

∣∣2

σ 2X

< p

)= 1

2Pkχ

2ν (p) , (4)

Fig. 3. Cross wavelet transform of the standardized AO and BMItime series. The 5% significance level against red noise is shownas a thick contour. The relative phase relationship is shown asarrows (with in-phase pointing right, anti-phase pointing left, andBMI leading AO by 90◦ pointing straight down).

where ν is equal to 1 for real and 2 for complex wavelets.The CWT of the AO and BMI are shown in Figs. 1 and 2

respectively. There are clearly common features in thewavelet power of the two time series such as the significantpeak in the ∼5 year band around 1940. Both series alsohave high power in the 2–7 year band in the period from1860–1900, though for AO the power is not above the 5%significance level. However, the similarity between the por-trayed patterns in this period is quite low and it is thereforehard to tell if it is merely a coincidence. The cross wavelettransform helps in this regard.

3.2 The cross wavelet transform

The cross wavelet transform (XWT) of two time series xn

and yn is defined as WXY =WXWY∗, where * denotes com-plex conjugation. We further define the cross wavelet poweras |WXY |. The complex argument arg(Wxy) can be inter-preted as the local relative phase between xn and yn in timefrequency space. The theoretical distribution of the crosswavelet power of two time series with background powerspectra P X

k and P Yk is given in Torrence and Compo (1998)

as

D

(∣∣WXn (s)WY∗

n (s)∣∣

σXσY

< p

)= Zν(p)

ν

√P X

k P Yk , (5)

where Zν(p) is the confidence level associated with the prob-ability p for a pdf defined by the square root of the product oftwo χ2 distributions. For example the 5% significance levelmarked in Fig. 3 is calculated using Z2(95%)=3.999.

The XWT of AO and BMI is shown in Fig. 4. Here wenotice that the common features we found by eye from theindividual wavelet transforms stand out as being significantat the 5% level. We also note that there also is significantcommon power in the ∼10–16 year band from 1940–1980.

564 A. Grinsted et al.: Cross wavelet and wavelet coherence

2 4 8 16 320.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Scale (s)

R2 5

% s

igni

fican

ce le

vel

AR1=[0.0 0.0],s/o=10AR1=[0.5 0.5],s/o=10AR1=[0.0 0.9],s/o=10AR1=[0.9 0.9],s/o=10AR1=[0.5 0.5],s/o=4

Fig. 4. Wavelet coherence 5% significance level determined usingMonte Carlo generated noise (with 10 000 surrogate data set pairs).The legend shows the 2 AR1 coefficients of the surrogate data setsand the numbers of scales per octave (s/o) used in calculating thescale smoothing. The color of the noise has little impact on thesignificance level, whereas the specifics of the smoothing have alarge impact. The large values at either end of the spectrum are dueto the scale smoothing operator reaching the scale boundaries.

For there to be a simple cause and effect relationship betweenthe phenomena recorded in the time series we would expectthat the oscillations are phase locked. So, it is comfortingthat the XWT show that AO and BMI are in anti phase inall the sectors with significant common power. Since AOand BMI are in anti-phase across all scales, we conclude thatBMI to a large extent simply mirrors the AO. Outside the ar-eas with significant power the phase relationship is also pre-dominantly anti phase. We therefore speculate that there is astronger link between AO and BMI than that implied by thecross wavelet power.

3.3 Cross wavelet phase angle

As we are interested in the phase difference between thecomponents of the two time series we need to estimate themean and confidence interval of the phase difference. Weuse the circular mean of the phase over regions with higherthan 5% statistical significance that are outside the COI toquantify the phase relationship. This is a useful and generalmethod for calculating the mean phase. The circular mean ofa set of angles (ai , i=1...n) is defined as (e.g. Zar, 1999)

am= arg (X, Y ) with X=n∑

i=1

cos(ai) and Y=n∑

i=1

sin(ai) , (6)

It is difficult to calculate the confidence interval of the meanangle reliably since the phase angles are not independent.The number of angles used in the calculation can be setarbitrarily high simply by increasing the scale resolution.However, it is interesting to know the scatter of angles around

the mean. For this we define the circular standard deviationas

s = √−2 ln(R/n) , (7)

where R=√(X2+Y 2). The circular standard deviation is

analogous to the linear standard deviation in that it variesfrom zero to infinity. It gives similar results to the linear stan-dard deviation when the angles are distributed closely aroundthe mean angle. In some cases there might be reasons forcalculating the mean phase angle for each scale, and then thephase angle can be quantified as a number of years.

The XWT phase angle within the 5% significant regionsand outside the COI has the mean phase –176±12◦ (where± designates the circular standard deviation). This basicallyconfirms the conclusion that AO and BMI are in anti-phase.Note that the time series already have a 3 month lag sincethe ice extent is greatest in April. A 3 month lag is consis-tent with the mechanism of stratospheric forcing of the tropo-sphere (Baldwin et al., 2001; Jevrejeva et al., 2003). The ob-servation that the phase angles are constant across all scalesargues for a constant time lag due to the physical mechanismof signal propagation from the AO to the ice extent. The de-viation from completely anti-phase suggests that AO leadsBMI slightly, however, the circular standard deviation is toolarge to make any firm conclusion.

3.4 Wavelet coherence

Cross wavelet power reveals areas with high common power.Another useful measure is how coherent the cross wavelettransform is in time frequency space. Following Torrenceand Webster (1998) we define the wavelet coherence of twotime series as

R2n(s) =

∣∣S (s−1WXY

n (s))∣∣2

S(s−1

∣∣WXn (s)

∣∣2)

· S(s−1

∣∣WYn (s)

∣∣2) , (8)

where S is a smoothing operator. Notice that this definitionclosely resembles that of a traditional correlation coefficient,and it is useful to think of the wavelet coherence as a lo-calized correlation coefficient in time frequency space. Wewrite the smoothing operator S as

S(W) = Sscale(Stime(Wn(s))) , (9)

where Sscale denotes smoothing along the wavelet scale axisand Stime smoothing in time. It is natural to design thesmoothing operator so that it has a similar footprint as thewavelet used. For the Morlet wavelet a suitable smoothingoperator is given by Torrence and Webster (1998)

Stime(W)|s =(

Wn(s) ∗ c

−t2

2s2

1

) ∣∣∣∣∣s

,

Stime(W)|s = (Wn(s) ∗ c2 (0.6 s))∣∣n, (10)

where c1 and c2 are normalization constants and is therectangle function. The factor of 0.6 is the empirically deter-mined scale decorrelation length for the Morlet wavelet (Tor-rence and Compo, 1998). In practice both convolutions are

A. Grinsted et al.: Cross wavelet and wavelet coherence 565

done discretely and therefore the normalization coefficientsare determined numerically.

The statistical significance level of the wavelet coherenceis estimated using Monte Carlo methods. We generate a largeensemble of surrogate data set pairs with the same AR1 co-efficients as the input datasets. For each pair we calculate thewavelet coherence. We then estimate the significance levelfor each scale using only values outside the COI. Empiricaltesting shows that the AR1 coefficients have little impact onthe significance level. The specifics of the smoothing opera-tor, however, have a large impact. For example the resolutionchosen when calculating the scale smoothing has a major in-fluence on the significance level (see Fig. 4). The MonteCarlo estimation of the significance level requires of the or-der of 1000 surrogate data set pairs. The number of scales peroctave should be high enough to capture the rectangle shapeof the scale smoothing operator while minimizing computingtime. Empirically we find 10 scales per octave to be satisfac-tory.

The squared WTC of AO and BMI is shown in Fig. 5.Compared with the XWT a larger section stands out as beingsignificant and all these areas show an anti-phase relation-ship between AO and BMI. The area of a time frequency plotabove the 5% significance level is not a reliable indicationof causality. Even if the scales were appropriately weighedfor the averaging, it is possible for two series to be perfectlycorrelated at one specific scale while the area of significantcorrelation is much less than 5%. However, the significant re-gion of Fig. 5 is so extensive that it is very unlikely that this issimply by chance. Oscillations in AO are manifested in theBMI on wavelengths varying from 2–20 years, suggestingthat BMI passively mirrors AO. Regions with low coherencecoincide with low wavelet power in the AO and are thereforeexpected. Possibly because the EOF that AO represents isnot really capturing the actual location of the centers of ac-tion during the Little Ice Age. As with the XWT, we can cal-culate the mean phase angle of the significant regions. Themean phase angle over the regions with significant waveletcoherence and outside the COI is 174±15◦.

4 Summary

The CWT expands a time series into a time frequency spacewhere oscillations can be seen in a highly intuitive way. TheMorlet wavelet (with ω0=6) is a good choice when usingwavelets for feature extraction purposes, because it is reason-ably localized in both time and frequency. From the CWTsof two time series one can construct the XWT. The XWT ex-poses regions with high common power and further revealsinformation about the phase relationship. If the two seriesare physically related we would expect a consistent or slowlyvarying phase lag that can be tested against mechanistic mod-els of the physical process. WTC can be thought of as the lo-cal correlation between two CWTs. In this way locally phaselocked behavior is uncovered. The more desirable featuresof the WTC come at the price of being slightly less localized

Fig. 5. Squared wavelet coherence between the standardized AOand BMI time series. The 5% significance level against red noise isshown as a thick contour. All significant sections show anti-phasebehavior.

in time frequency space. The significance level of the WTChas to be determined using Monte Carlo methods.

4.1 Practical tips

Cross wavelet analysis and wavelet coherence are powerfulmethods for testing proposed linkages between two time se-ries.

– Check the histograms of the time series to ensure thatthey are not too far from normally distributed. Considertransforming the time series, if the pdf’s of the time se-ries are far from Gaussian. When choosing a transfor-mation, it is preferable to choose an analytic transfor-mation such as taking the logarithm if the data is log-normally distributed. In other cases the simple “per-centile” transformation we used for the BMI might beuseful. An advantage of using that particular transfor-mation is that it does not have any outliers.

– Consider what the expectations are for the outcome ofthe analysis given the proposed linking mechanism. Wecaution against blindly applying these methods to ran-domly chosen data sets. Like other statistical tests somedata set sets will display highly statistically significantlinks simply by chance.

– When a wavelet has been chosen the CWTs of both timeseries are calculated. We suggest a scale resolution of10 scales per octave and use of the Morlet wavelet un-less there are good grounds to do otherwise. For geo-physical time series an AR1 red noise assumption is of-ten suitable and Eqs. (3) and (4) can be used to calculatethe significance level of the wavelet power. Rememberto take special care not to misinterpret results inside theCOI.

566 A. Grinsted et al.: Cross wavelet and wavelet coherence

– From the two CWTs the XWT is calculated. The XWTexposes regions with high common power and furtherreveals information about the phase relationship. If thetwo series are physically related we would expect a con-sistent or slowly varying phase lag that can be testedagainst mechanistic models of the physical process. Thecircular mean of the phase angles can be used to quan-tify the phase relationship.

– Also, from two CWTs the WTC can be calculated whichcan be thought of as the local correlation between thetime series in time frequency space. Where XWT un-veils high common power, WTC finds locally phaselocked behavior. The more desirable features of theWTC come at the price of being slightly less localizedin time frequency space. The significance level of theWTC has to be determined using Monte Carlo methods.

A MatLab software package by the authors for performingXWT and WTC can be found at http://www.pol.ac.uk/home/research/waveletcoherence/.

Acknowledgements. Financial assistance was provided by theThule Institute. Some of our software includes code originallywritten by C. Torrence and G. Compo that is available at:http://paos.colorado.edu/research/wavelets/ and by E. Breiten-berger of the University of Alaska which were adapted from thefreeware SSA-MTM Toolkit: http://www.atmos.ucla.edu/tcd/ssa.

Edited by: M. ThielReviewed by: two referees

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Omstedt, A. and Chen, D.: Influence of atmospheric circulation onthe maximum ice extent in the Baltic Sea, J. Geophys. Res., Vol.106, 4493–4500, 2001.

Seina, A., Gronvall, H., Kalliosaari, S., and Vainio, J.: Ice seasons1996–2000 in Finnish sea areas / Jaatalvet 1996–2000 Suomenmerialueilla, Meri, Report Series of the Finnish Institute of Ma-rine Research, 43, 2001.

Thompson, D. W. J. and Wallace, J. M.: The Arctic Oscillation sig-nature in the winter geopotential height and temperature fields,Geophys. Res. Lett., 25, 1297–1300, 1998.

Torrence, C. and Compo, G. P.: A practical guide to wavelet analy-sis, Bull. Am. Meteorol. Soc., 79, 61–78, 1998.

Torrence, C. and Webster, P.: Interdecadal Changes in the ESNO-Monsoon System, J. Clim., 12, 2679–2690, 1999.

Zar, J. H.: Biostatistical Analysis, Prentice hall, New Jersey, 1999.

III

Svalbard summer melting, continentality, and sea ice extent from the

Lomonosovfonna ice core

Aslak Grinsted,1,2 John C. Moore,1 Veijo Pohjola,3 Tonu Martma,4 and Elisabeth Isaksson5

Received 13 July 2005; revised 25 October 2005; accepted 18 January 2006; published 14 April 2006.

[1] We develop a continentality proxy (1600–1930) based on amplitudes of the annualsignal in oxygen isotopes in an ice core. We show via modeling that by using 5 and15 year average amplitudes the effects of diffusion and varying layer thickness can beminimized, such that amplitudes then reflect real seasonal changes in d18O under theinfluence of melt. A model of chemical fractionation in ice based on differing elution ratesfor pairs of ions is developed as a proxy for summer melt (1130–1990). The best pairsare sodium with magnesium and potassium with chloride. The continentality and meltproxies are validated against twentieth-century instrumental records and longer historicalclimate proxies. In addition to summer temperature, the melt proxy also appears to reflectsea ice extent, likely as a result of sodium chloride fractionation in the oceanic sea icemargin source area that is dependent on winter temperatures. We show that the climatehistory they depict is consistent with what we see from isotopic paleothermometry.Continentality was greatest during the Little Ice Age but decreased around 1870, 20–30 years before the rise in temperatures indicated by the d18O profile. The degree ofsummer melt was significantly larger during the period 1130–1300 than in the 1990s.

Citation: Grinsted, A., J. C. Moore, V. Pohjola, T. Martma, and E. Isaksson (2006), Svalbard summer melting, continentality, and sea

ice extent from the Lomonosovfonna ice core, J. Geophys. Res., 111, D07110, doi:10.1029/2005JD006494.

1. Introduction

[2] Ice cores hold many potential proxies of climate. Thedifficulty is usually to assign a proxy to a particular climaticvariable [Jones and Mann, 2004]. One of the most wellknown is d18O, which is the deviation of the isotopic ratiobetween H2O

18 to H2O16 from the ratio in standard seawater

[Dansgaard, 1964]. In precipitation d18O is generally ac-cepted as being a proxy for condensation temperature in theatmosphere and is related to the surface temperature inmodern snow [van Lipzig et al., 2002].[3] However, the relationship between d18O and surface

temperature changes under different climate regimes as it isaffected by changes in moisture source temperature, sea-sonality of precipitation and moisture transport path. Inaddition to mean temperature, the seasonal difference be-tween summer and winter is of broad environmental signif-icance. Continental interiors have larger seasonal temperaturevariations than maritime areas and the annual temperaturerange is often used as a measure of continentality. Ice coresfrom the interiors of large ice sheets such as Greenland orAntarctica are probably not very sensitive to the location ofthe sea ice margin, which determines the distance to open

water and hence influences continentality, as it is far awayfrom the drilling site. However, an ice core froman island nearthe seasonal sea ice margin may reflect the much largerrelative variations in distance to open water via a continen-tality index. In ice cores where the seasonal cycle in d18O iswell preserved the amplitude is an obvious candidate as aproxy for continentality. Unfortunately the oxygen isotoperecord is often affected by diffusion on a scale comparable tothe annual layer thickness. It is relatively common toback diffuse the isotopic profile to amplify the annualcycle for layer counting purposes [Bolzan and Pohjola,2000]. However, the reconstructed amplitude from backdiffusion is very sensitive to small errors [Johnsen et al.,2000] and, in sites with seasonal melt, back diffusion isproblematic as it is expected that diffusion acts veryunevenly.[4] Stratigraphic melt index (SMI) is defined as the

weight fraction of clear bubble-free ice. SMI has been usedas a proxy for summer temperature on Canadian ice caps[e.g., Koerner, 1997]. However, in Greenland, Pfeffer andHumphrey [1998] suggest that melt layers not only reflectwarm summer conditions but also the temperature gradientexperienced by the snowpack during the preceding months.Therefore, under some conditions, SMI is also affectedby the seasonal temperature range, or continentality, andaccumulation rate.[5] Almost all ice cores in the Arctic outside of Greenland

are influenced by seasonal melt. Meltwater can refreeze onthe surface, percolate into deeper layers or run off. Ifmeltwater refreezes in the surface layers, then it has littleimpact on average ion concentrations and d18O values.

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, D07110, doi:10.1029/2005JD006494, 2006

1Arctic Centre, University of Lapland, Rovaniemi, Finland.2Department of Geophysics, University of Oulu, Oulu, Finland.3Department of Earth Sciences, Uppsala University, Uppsala, Sweden.4Institute of Geology, Tallinn University of Technology, Tallinn,

Estonia.5Norwegian Polar Institute, Tromsø, Norway.

Copyright 2006 by the American Geophysical Union.0148-0227/06/2005JD006494$09.00

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Percolation and runoff affects ions much more strongly thanisotopes because most ions have a very strong affinity forthe water phase and even small amounts of meltwater willhave high concentrations of ions [Davies et al., 1982].Percolation however only redistributes the ions in thesnowpack whereas they are lost by runoff. Iizuka et al.[2002] use the preferential removal of ions from particularlayers as an indicator for seasonal melt on Austfonna,Svalbard, and find empirically that the magnesium sodiumratio was the best melt proxy.[6] In 1997 we drilled a 121-m-long ice core (spanning

about 800 years) on Lomonosovfonna [Isaksson et al.,2001], the highest ice field in Svalbard (78�5105300N,17�2503000E, 1255 m above sea level (asl)). Total ice depthfrom radar sounding was 123 m, and the site is close to thehighest point of the ice cap with roughly radial ice flow.Dating of the core was based on a layer-thinning model tiedwith the known dates of prominent reference horizons (seeKekonen et al. [2005] for details). There is evidence thatd18O is a good proxy for surface air temperature at the drillsite: the rise in d18O from 1900 to 1920 that mirrors the risein temperature seen in the instrumental record from Sval-bard [Isaksson et al., 2003], consistency with the boreholetemperature profile [van de Wal et al., 2002], and well-preserved annual cycles [Pohjola et al., 2002b]. The currentannual temperature range is from 0�C to about �40�C,though temperatures within the firn pack tend to be muchwarmer, approaching �3�C at 15 m depth [van de Wal etal., 2002].[7] The soluble ion chemical record in the ice core spans

the last 800 years [Kekonen et al., 2005] and shows clearlydefined changes corresponding with different climatic peri-ods and anthropogenic pollution. For the Lomonosovfonnacore the impact of percolation is much less significant thanthe impact of changes in precipitation chemistry [Moore etal., 2005]. All ions except ammonium have anomalouslylow concentrations in the oldest ice of the core. This isinterpreted as loss of ions in the original snowpack causedby warm conditions prior to about 1300 [Kekonen et al.,2005]. SMI for the core shows no relationship with oxygenisotopes [Pohjola et al., 2002a], and thus we cannotinterpret SMI as a proxy for summer temperature at the site.[8] In this paper we develop the theoretical basis for both

continentality and melt proxies and apply them to the chem-ical [Kekonen et al., 2005] and the oxygen isotope records[Isaksson et al., 2003] from the Lomonosovfonna ice core.We test the ice core proxies against instrumental temperatureand historical sea ice extent over the last 150 years and thenuse the ice core proxies to derive climatic variables back to1130. While summer temperature proxies are common, suchas tree ring width chronologies, we will show that we haveproxies of both summer melting and annual temperaturerange, both of which seem to be influenced by regional seaice extent.

2. Data

[9] The oxygen isotopic record (d18O) has been measuredat 5 cm resolution [Isaksson et al., 2003]. The concentrationof ions (Cl�, NO3

�, SO42�, CH3SO3

�, NH4+, K+, Ca2+, Mg2+,

Na+) has been measured in 5 cm samples taken at 10 cmintervals along the whole core. Sample recovery and anal-

ysis are discussed by Kekonen et al. [2005, and referencestherein]. SMI has been measured at 5 mm resolution in thetop 80 m (1707–1997 A.D.) of the core by visuallyclassifying into firn, bubbly ice, diffuse ice and clear iceand assigning a melt percentage to each category [Pohjolaet al., 2002a].[10] We will compare our ice core proxies to local

historical temperatures and also to Barents Sea ice extentsince this is a major driving force of regional climate[Isaksson et al., 2003, 2005a, 2005b]. Svalbard Airportreconstructed temperature, from 80km south west of thedrill site, is the closest instrumental historical record. This isa composite record from sites near Longyearbyen and hasmonthly mean temperatures since 1911 [Nordli et al., 1996].We extract winter (TDJF), summer (TJJA) and yearly meantemperatures (Tm) as well as yearly temperature range (DT).Records of Barents sea ice extent have been compiled fromship logbooks [Vinje, 2001]. April sea ice extent for theeastern and western Barents Sea (Be and Bw, respectively)covers the period from 1864 to 1999, with few gaps mostnotably in the WWII years. We have left the missing yearsout of correlations with other records.

3. Methods

3.1. Continentality From D18O

3.1.1. Factors Contributing to D18O Amplitudes[11] The amplitude of the annual signal in d18O (A) is an

obvious candidate for a continentality proxy, since d18O onLomonosovfonna is a proxy for surface air temperature[Isaksson et al., 2001; van de Wal et al., 2002] and theannual cycle is well preserved [Pohjola et al., 2002b].However, there are complicating factors in using the mea-sured amplitudes. Meltwater percolation redistributes iso-topes, effectively reducing the annual amplitudes. Further,the amplitudes decay with depth because of isotopic diffu-sion [Johnsen et al., 2000].[12] Diffusion acts so as to smooth out steep gradients in

the signal, so thin layers with a given initial amplitude ofd18O are smoothed faster than thick layers with the sameinitial amplitude. Therefore one could argue that A is aproxy for accumulation rate rather than for continentality. Inthe firn layer (top 30 m on Lomonosovfonna) diffusion isvery fast because of vapor transport, and therefore ampli-tudes decay rapidly in this layer. Solid ice diffusion is amuch slower process than firn diffusion. The modeledamplitudes would change only about 5% between thebottom of the firn layer (defined as the depth were poresare closed off from the atmosphere) and AD1600 (usingdiffusion rates from Johnsen et al. [2000]), and so solid icediffusion can be ignored. Thin ice layers formed duringmelt/refreeze events will act as vapor barriers and limit firndiffusion. To ensure we are below the effect of vaportransport, we restrict ourselves to the record below 30 m.That means we can reasonably assume that there is verylittle trend in the characteristic diffusion length scale in thispart of the core, though there will be local variability (e.g.,due to ice layers).[13] Previous studies on Lomonosovfonna have found

that percolation lengths are generally shorter than the annuallayer thickness [Pohjola et al., 2002a] except during thewarmest years (such as post-1990) where percolation length

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appear to be 2–8 annual layers [Kekonen et al., 2005]. Fromsnow pit studies carried out in May 2002 on Lomonosov-fonna we find that the annual isotopic range prior to melt isapproximately 10%. We consider now the scenario where afraction of the summer snowmelts, percolates and thenrefreezes in the winter snow. Prior to melt we assume thatd18Osummer = �10% and d18Owinter = �20%. Isotopicfractionation is observed to occur during freezing of ice,where an isotopic gradient will form in the ice layer becausethe first ice that forms will be relatively enriched in H2

18Obecause of Rayleigh fractionation [e.g., Hubbard andSharp, 1993]. The ice will be �3% heavier than the waterit freezes from. The same effect is not seen during meltingand the meltwater has the same isotopic content as thesummer snow (d18Owater = �10%) because the ice inindividual crystals is effectively unmixed. The refrozenmeltwater will therefore have average d18O of �10% withthe heaviest fraction having �7%. However, it will havebeen mixed with a fraction of winter snow. In the casewhere the refrozen layer composition originates from onethird meltwater and two thirds winter snow (equivalent to�50% SMI) the resulting layer will have mean d18Orefrozen =�16.7% with the heaviest being �15.7%. The net effect isthat the annual isotopic range will have been reduced from10% to �6% because of percolation, which compares wellwith the isotopic values in the upper firn pack on Lomo-nosovfonna [Pohjola et al., 2002a]. However, thin ice layerswill not cause any significant smoothing and could eveninhibit firn diffusion. For this reason we may expect that theamplitudes respond nonlinearly to SMI. Surface melt willgenerally dampen the signal whereas limited melt inhibitssmoothing by firn diffusion. Observational data (Figure 4)will be discussed later.3.1.2. Accumulation Rate Effects on Amplitudes[14] Rather than attempting the difficult back diffusion

problem, which is sensitive to small errors in diffusionlength, we argue that a statistical mean over several yearswill minimize the integrated effects of diffusion. That is, wewill assume that adequate time smoothing will give a recordthat is close to proportional to the annual amplitude infreshly precipitated snow (A0). To test this assumption, we

will apply a mathematical diffusion model to an artificialrecord of amplitudes with realistically distributed layerthickness. From this we will give an estimate of how muchof the total variance in A is due to diffusion processes and sohow much smoothing is needed. We define s as thecharacteristic diffusion length scale [Johnsen et al., 2000].The observed layer thickness (l) has a wide distribution(Figure 1), varying by a factor of 5 or more, while variationsin s due to ice layers are, we argue, much smaller; we willreturn to this point later. The ratio l/s, which measures theannual layer scale to the diffusion length scale, will bedominated by changes in l and we will assume a constant sin the development that follows.[15] The net effect of diffusion is equivalent to convolut-

ing the deposited signal with a Gaussian filter having astandard deviation s (which is also the diffusion length),[Johnsen et al., 2000]. Using the Fourier convolutiontheorem, the power spectral density of the isotopic profileis then

P Fð Þ ¼ P0 Fð Þe� 2pFsð Þ2 ; ð1Þ

where F is the frequency and P0 is the spectral density of theinitial deposited d18O profile, which is, however, com-pressed by layer thinning with depth due to ice flow. Wewill estimate P0 from the near surface layers and thenestimate s by fitting equation (1) to the measured P from asection of the profile just below pore close off. If we assumethat the isotopes are deposited with a red noise (AR1)background spectrum with lag-1 autocorrelation a, we canwrite

Psurf Fð Þ ¼ 2Vdz1� a2

1� ae�2ipFdzj j2 ; ð2Þ

where dz is the sampling resolution and V is the variance ofthe deposited signal [Torrence and Compo, 1998]. We willuse water equivalent (w.e.) depth units to minimize layerthinning with depth, and therefore be less sensitive toprecise determination of pore close off depth. We estimatethe deposited signal to have a = 0.85 from the top 4 m w.e.of the core, sampled at a resolution of 0.03 m w.e. At depth,the background spectrum will have been both whitenedbecause of compression and reddened because of diffusion.We assume that P0 is proportional to Psurf with thefrequency axis stretched to account for thinner layers atdepth.We can then write P0(F) = cPsurf(rF), where r accountsfor spectrum whitening and c represents the difference inisotopic variance between the present and at depth. Wedetermine the diffusion length by fitting equation (1) (with anadditional white measurement noise term) to the observedpower spectrum using the maximum entropy method [Chen,1988]. Themeasurement noise is estimated to have a standarddeviation of 0.06% from the high-frequency variability of theoldest and most diffused ice. This is consistent with anestimatedmeasurement precision of ±0.1% based on analysisof replica samples. In Figure 2 the diffusion length has beendetermined through minimizing the mean absolute deviationof the theoretical spectrum to the observed in a movingwindow and has then been decompressed to pore closeoff level. The variation in diffusion length observed in

Figure 1. Histogram of layer thickness from 25 to 37 m(i.e., near pore close off) and the lognormal fit used in themodeling.

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Figure 2 increases back in time and does not correlatewith SMI nor is there any hint of a nonlinear relationshipin a scatterplot. Therefore the variation is more likelycaused by uncertainties in the estimated spectrum due totoo few samples in the short window or becausecompression makes it harder to resolve the diffusionlength with the given sampling resolution, rather thanactual variations in diffusion length. The fact that there ispractically no trend in the diffusion length providesjustification for the assumption that solid ice diffusion canbe ignored.[16] As a consequence of equation (1), the amplitude of

the annual signal can be written as

A ¼ A0e�2 ps

lð Þ2 ; ð3Þ

where A0 is the initial amplitude and l is the compressedlayer thickness. Measured l [Pohjola et al., 2002b] is closeto lognormally distributed (Figure 1). The log10(l) hasmean �0.67 and standard deviation 0.22 at pore close off.Using these statistics, we generate a long surrogate timeseries of l and calculate how a diffusion length s = 0.030 mw.e. affects the amplitudes. The result can be seen inFigure 3. At 30 m depth, A has on average decayed to 61%of the deposited A0, but its distribution is very wide. Table 1shows that 5% of the layers will have decayed to less than13% of the initial value, while another 5% of layers onlydecayed by 7%. However, taking 15 year moving averagesof A tightens the distribution significantly (Figure 3). The

proportion of variance (W) in A due to variation in l can beestimated as

W ¼A2var e�2 pslð Þ2

� �

var Að Þ ; ð4Þ

where A is the mean value of A around pore close off. Usingthe variances from Table 1, we find that with 5 and 15 yearmoving averages only 11% and 7% of the total variance inA is due to variations in l, so we can say that A is nearlyproportional to A0. That is, adequate time smoothing willgive a statistically representative sampling of both thin andthick layers. In a similar fashion we determine that a 25%change in s causes a �10% change in amplitudes. Ourhypothesis is therefore that smoothed A on Lomonosovfon-na reflects real seasonal changes in d18O influenced by meltrather than changes in accumulation rate or diffusion length.We will test this hypothesis in the results section.3.1.3. Amplitude Reconstruction[17] To derive the annual amplitudes record, we first up-

sampled the isotope record to 10 samples per year, and thenhigh-pass filtered it with a cutoff wavelength of 3 years. Theamplitude was then measured as the range in a window 1 yearwide from the modeled timescale [Kekonen et al., 2005].This processing can be carried out beyond the limits oflayer counting because measuring the amplitude is sub-stantially easier than counting the layers. Even if thisprocedure misses a peak and picks the one from aneighboring year, the amplitude will still be a reasonableestimate. We tested our results by varying the windowlength and the high-pass-filtering method, and found themto be robust.[18] The amplitudes decay as the yearly layer thickness is

compressed by ice flow, eventually to a thickness compa-rable with the 5 cm sampling resolution of d18O. The effectof the sampling is equivalent to convoluting the ‘‘true’’

Figure 2. Best fitting diffusion length decompressed topore close-off level, obtained by minimizing the meanabsolute deviation between the theoretical power spectrum(equation (1)) and the low-order maximum entropy powerspectrum estimated in 15 year (thin line) and 60 year (thickline) windows. The curve based on 15 year windows is cutoff at 1740 because there are too few samples in the windowto accurately estimate the power spectrum. Similarly, thegap near 1700 is due to discontinuous sampling. Thevariability in sco increases back in time as the number ofsamples in the window decreases and as compression makesit problematic to resolve the diffusion length with the givensampling resolution. This suggests that the variability reflectsfitting uncertainty rather than a climate signal.

Figure 3. (top) Histogram of modeled annual amplitudesat pore close off relative to deposited amplitudes assuming adiffusion length of 3.0 cm w.e. (middle and bottom) Using 5and 15 year moving averages (middle and bottom) tightensand normalizes the distribution significantly.

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signal with a rectangle function. Hence, using the Fourierconvolution theorem, we correct the amplitudes by applying

Acorrected ¼ Ameasured

sinc p Dzl

� � ð5Þ

where Dz is the width of a sample and l is the localwavelength of the annual signal. At 1700 the correction factoris 1.3, while at 1600 the factor we apply is 1.6. Somewhatarbitrarily, we decide that beyond 1600 the errors in thecorrection factor are too big to extract a meaningful signal.

3.2. Chemical Washout Index

[19] Many studies have found that particular ions arepreferentially washed out of the ice during seasonal melt[e.g., Davies et al., 1982; Iizuka et al., 2002]. However,details of specific ion elution rates depend on the incorpo-ration of ions in the ice during grain growth and so arecertainly affected by processes that occur prior to the onsetof melting [Cragin et al., 1996]. While microphysicalprocesses governing the removal of ions are poorly under-stood, it is reasonable to assume that the ions are removedfractionally as the melt progresses, and so we can write

@C

@W¼ �a � C and C 0ð Þ ¼ Cp , C Wð Þ ¼ Cp � e�aW ; ð6Þ

where W is a dimensionless washout index, C is theconcentration of a chemical species, a is a species-specificdimensionless washout rate and Cp is the species initialconcentration in the fresh snow. Here we have implicitlyassumed that the reservoir volume for the ions is constant;that is, the increase in concentration due to a reducedvolume caused by runoff is negligible, which is differentfrom the situation in a seasonal snowpack.[20] Some chemical ions have the same principle source.

For example, both Na+ and Mg2+ are mainly of sea saltorigin in Svalbard snow [Virkkunen, 2004; Iizuka et al.,2002], and so their concentrations are often taken to belinearly dependent. If we define X and Y to be the concen-trations of two such chemical species, then Xp = kxyYp,where Xp and Yp are the concentrations of X and Y prior tomelt, and kXY is their ratio in fresh snow. From equation (6)it is clear that the logarithmic ratio of the concentrationsafter washout is linear in W.

logX

Y

� �¼ log

Xp � e�aXW

Yp � e�aYW

� �¼ log kXYð Þ þ aY � aXð ÞW : ð7Þ

[21] Hence we can derive a washout index provided thatthe washout rates aX and aY differ significantly. Note that

equation (7) holds even for a finite reservoir that is depletedduring the melt season, as happens with significant runoff.Iizuka et al. [2002] show that there are high correlations(>0.95) between the concentrations of Na+, Cl�, K+ andMg2+ in Svalbard dry snow from Austfonna. Virkkunen[2004] find similar results from Lomonosovfonna, indicat-ing that these ions have the same (marine) source forSvalbard, justifying the use of equation (7). However, allthese correlations break down in wet snow, except thatbetween Na+ and Cl�. Hence the washout rates for Na+, K+

and Mg2+ must differ widely, whereas they are the similarfor Na+ and Cl�. In particular, Iizuka et al. [2002] find thatMg2+ is washed out more efficiently than Na+ (aMg > aNa).Because of these differences in washout rates, we canconstruct two independent chemical melt indices WNaMg

and WClK based on log(Na+/Mg2+) and log(Cl�/K+). Weconsider WNaMg to be superior to WClK since relativemeasurement errors in K+ are much larger than in Mg2+

[Kekonen et al., 2002] and further K+ has a tendency tohave large peaks that are seemingly unrelated to other ionsin high-resolution pit data [Iizuka et al., 2002]. Usingseawater ratios in mgL�1, we get kNaMg = 8.3 andkClK = 49. To avoid difficulties with percolation betweenlayers [Pohjola et al., 2002a; Moore et al., 2005], we use15 year mean concentrations. To compare WNaMg and WClK,we scale the dimensionless indices by their mean valuesover the twentieth century.

4. Results

[22] We test the new indices by comparison with theinstrumental record and proxy data. Isaksson et al. [2005b]compare the d18O profile with meteorological records fromaround the Barents Sea with Vardø on the Norwegian coasthaving the best correlation. However, the Vardø temperaturerecord is longer than that from Longyearbyen, and thereforeit is expected that it will correlate better simply because itcovers the large increase in temperatures at the end of theLittle Ice Age (LIA). Over the period of common overlap(1911–1997), d18O correlate better with Longyearbyen thanVardø temperatures. Therefore our interpretation is thatLongyearbyen temperatures are more representative of theconditions on Lomonosovfonna than those from Vardø.[23] Correlation coefficients between 5 year moving

averages of ice core data and the historical records are shownin Table 2. As expected isotopes (d18O) correlate significantlywith Longyearbyen temperatures (Tm, TDJF, TJJA). To avoidthe problems with firn diffusion completely, we shoulddiscard the continentality record above pore close off (e.g.,younger than 1930). However, to test A against instrumentalrecords we need a bigger overlap and so, as a compromise, wechoose to truncate A at 1970 (15.6 m depth) by whenmost of the firn diffusion already has taken place. Themore continental the climate conditions, the greater thetemperature range with winter temperatures changing themost. So, as expected, the continentality proxy (A)correlates with temperature range (DT), while d18O anti-correlates (Table 2). On longer timescales we find that Ais also highly anticorrelated with d18O (Table 3). This isconsistent with the interpretation of d18O as a temperatureproxy and A as a continentality proxy. We find no clearrelationship between A and SMI (Figure 4) and we

Table 1. Percentiles and Variance of Modeled Annual Isotopic

Amplitudes (Equation (3)) After Firn Diffusion Relative to Their

Deposited Amplitudes Assuming That the Layer Thickness is

Lognormally Distributeda

Percentile

Variance5% 50% 95%

A/A0 0.12 0.67 0.93 0.066A5/A0 0.42 0.62 0.78 0.013A15/A0 0.51 0.62 0.72 0.005

aA5 and A15 denote 5 and 15 year moving averages, respectively.

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therefore conclude that melt is not a major contributor tothe variability in A.[24] Linear regression gives Tm = 1.12 d18O + 11.5 (R2 =

0.22) and DT = 3.3 A + 19.5 (R2 = 0.32), and both of theseregressions are significant at the 95% level. However, van deWal et al. [2002] analyzed the borehole temperature profilefinding that temperatures in the nineteenth century were2.4�C colder than the twentieth century, this compares witha 0.8% difference in d18O [Isaksson et al., 2003] between thenineteenth and twentieth centuries. This indicates that theabove regression underestimates the sensitivity of d18O ontemperature, which is to be expected considering that thehistorical records used in the regression only cover therelatively warm twentieth-century period [von Storch et al.,2004].[25] The SMI does not correlate significantly with tem-

peratures. We find that the full records of SMI and d18O aresignificantly anticorrelated (Table 3). Thus, if we believed18O to be a temperature proxy, it does not seem plausible toconsider SMI a summer temperature proxy. The alternativeexplanation of SMI as a continentality index can also berejected since there is no significant correlation with A(Table 3 and Figure 4). Perhaps the SMI, for this ice core,is a composite signal that requires a more sophisticatedtreatment of percolation effects.[26] Perhaps surprisingly, there is no significant correla-

tion between washout indices and any temperature series(Table 2). However, there are significant correlations withd18O. These can be explained by the local differences inconditions. The summit of Lomonosovfonna is inland and1250 m asl, whereas Longyearbyen is within a fjord at sealevel. In summer, warm days often produce foggy condi-tions near sea level, and sunny conditions at higher eleva-tions, therefore we expect that the washout indices arerecording local summer melt.

[27] It is noticeable that both washout indices have nega-tive values throughout extensive periods (Figure 5). If, asseems reasonable, runoff is the only postdepositional processthat can change 15 year averages of these ion ratios and giventhat runoff would act to make the indices more positive,negative values can only be explained by changing the ionicratios of the precipitation. Either additional sources accountfor the excessMg2+ and K+ or fractionation processes depleteNa+ and Cl� relative to Mg2+ and K+. Since there aresimultaneous excesses of both Mg2+ and K+ an additionalsource contributing to both ions, such as terrestrial dust(though it would need to be relatively depleted in Na+),would produce changes in both washout rates. However,Table 2 shows that the washout rates are significantlyinversely correlated with sea ice extent in the Barents Sea,which points toward fractionation processes as the mostlikely explanation. Rankin et al. [2002] proposed fraction-ation by frost flowers on sea ice to explain low ratios of SO4

2�

in Antarctic ice, but such depletion cannot be seen in theArctic data because of the large magnitude of non-sea-saltSO4

2� sources. However, the same mechanism could alsoremoveNaCl from source aerosol.Whenwinter temperaturesdrop below the NaCl eutectic (�22�C), frost flowers will berelatively enriched in ions that remain liquid at low temper-atures, including Mg2+ and K+. Direct evidence of depletedNa+/Mg2+ and Na+/K+ ratios on sea ice have been found byDomine et al. [2004], though they suggest that an additionalterrestrial source of K+ and Mg2+ is involved. Two indepen-dent effects influence the ionic ratios: increased runoff raisesthe ratios, while increased sea ice lowers them. Thus lowwashout indices indicate a cold climate with relatively moresea ice. Consistently, we find that WNaMg and WClK correlatewith d18O and anticorrelate with A (Table 3).

5. Discussion

[28] The Lomonosovfonna d18O record indicates that theLIA in Svalbard ended around 1900 (Figure 5). The start of

Table 2. Significant Correlation Coefficients (p < 0.05) Between

Ice Core Data and Historical Records for Their Overlapping

Periodsa

d18O A SMI WNaMg WClK

Tm 0.47 �0.38TJJA 0.52TDJF 0.53 �0.48DT �0.34 0.57Bw �0.65 �0.22 �0.28 �0.60Be �0.34 �0.34 �0.50 �0.32

aAll records have been smoothed using 5 year moving averages, so as tomake the results insensitive to dating errors. Ice core parameters: A,continentality; SMI, stratigraphic melt index; Wxy, chemical washoutindices. Historical parameters: T, Longyearbyen temperature (mean,summer, winter, annual range); B, April sea ice extent in the western andeastern Barents Sea.

Table 3. Significant Correlation Coefficients (p < 0.05) for

Lomonosovfonna Ice Core Dataa

d18O A SMI WNaMg WClK

d18O �0.66 �0.48 0.60 0.51A �0.59 �0.46 �0.46SMI �0.48 �0.16 �0.12WNaMg 0.60 �0.43 �0.16 0.73WClK 0.51 �0.36 �0.12 0.73

aParameters are as in Table 2. All records have been smoothed using15 year moving averages.

Figure 4. Scatterplot of annual amplitudes (A) againststratigraphic melt index (SMI) using 15 year movingaverages (1700–1970). There is no simple relationshipbetween melt and amplitudes. Scatter increases when using5 year averages.

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the LIA is less easy to define as there seems to be a gradualdescent into colder climatic conditions. However, by 1600 aclear signature of the LIA is seen in the isotopes andchemistry [Isaksson et al., 2003; Kekonen et al., 2005],probably indicating that the sea ice margin had advancedclose enough to Svalbard to affect local climate. Continen-tality and washout indices show more gradual changes. Thecoldest and most continental period seen in the core is thenineteenth century. For comparison, most western NorthAmerican proxies also show the nineteenth century as thecoldest period, whereas in European proxies the coldestperiod occurs between 1600 and 1700 [Jones and Mann,2004]. Under the reasonable assumption that sea salt ionshave the same source throughout the record, negativewashout indices during the period from 1700 to 1900 areindicative of greater sea ice extent in the Barents Sea.[29] The continentality proxy (A) peaks at 1860 and

declines rapidly thereafter. This suggests that a change inclimate was already setting in by 1870 and temperatures andsea ice extent lagged the continentality change by 30 years.However, continentality is most affected by winter temper-atures, while sea ice extent is at a maximum in spring. Thuswe suggest that winter temperatures rose several decadesbefore spring and summer temperatures.[30] In the oldest part of the core (1130–1200), the

washout indices are more than 4 times as high as thoseseen during the last century, indicating a high degree ofrunoff. Since 1997 we have performed regular snow pitstudies [Virkkunen, 2004], and the very warm 2001 summerresulted in similar loss of ions and washout ratios as theearliest part of the core. This suggests that the MedievalWarm Period [Jones and Mann, 2004] in Svalbard summerconditions were as warm (or warmer) as present-day,

consistent with the Northern Hemisphere temperature re-construction of Moberg et al. [2005].

6. Conclusion

[31] We hope that this paper inspires interest in usingArctic ice cores from smaller ice caps for reconstruction ofLate Holocene environmental history on a regional scale.We have developed proxies for the washout intensity(WNaMg and WClK) based on the ion ratios of sea salt ions(Na+, Cl�, Mg2+ and K+) and continentality (A) based onannual amplitudes of d18O. The washout indices can providea valuable constraint on whether ion data truly representsprecipitation chemistry or if runoff has affected the record.For Lomonosovfonna we find that sea ice extent also affectsthe washout indices, most likely because NaCl is fraction-ated relative to other salts in the source area. Luckily thesign of this fractionation is such that WNaMg and WClK stillcan be thought of as proxies for summer temperatures.Washout indices confirm results from earlier studies[Kekonen et al., 2005] showing that the ion chemistry in theearliest part of the core (before 1200) was dominated bywashout of ions. We find that washout intensities are morethan 4 times higher than the average over the last century.[32] Annual amplitudes in d18O are affected by melt and

diffusion in a nontrivial and likely nonlinear manner. In thecase of Lomonosovfonna, the d18O record is subject to meltand percolation and therefore presumably also to highlyanisotropic diffusion in the firn. Despite this, we show thatamplitudes below the level of strongly active firn diffusionare a good proxy for continentality at Lomonosovfonna, ifsufficiently smoothed. However, further studies on othercores are needed before amplitudes generally can be con-

Figure 5. Fifteen-year moving averages of Lomonosovfonna ice core data. (a) Oxygen isotopes,(b) continentality proxy (A), (c) stratigraphic melt indices (SMI), and (d) washout indices (solid line isWNaMg, and dashed line is WClK).

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sidered to be a proven proxy for continentality. The crucialstep is to smooth the amplitude record sufficiently so thateach value represents a statistical mean over several yearsminimizing variations in annual layer thickness. An impor-tant assumption is that the trends in diffusion length andaccumulation rate are small compared with the width of thelayer thickness distribution. If such trends are significant,their contribution to the total variation in A may beestimated using a straightforward modification of the modelpresented in this paper. Under some conditions one mayargue that continentality is constant (e.g., Central Greenlandover the Holocene) and A might be used to infer changes indiffusion length.[33] On Lomonosovfonna we find that the end of the LIA

is very well defined in d18O and agrees with the earliest risein washout indices (WNaMg, WClK) around 1880. The Con-tinentality proxy (A), however, starts to decline after 1860.We suggest that the response of spring and summer temper-atures and the decline in sea ice in the Barents Sea laggedthe rise in winter temperatures by several decades.

[34] Acknowledgments. The drilling of the Lomonosovfonna 1997ice core was financed by the Norwegian Polar Institute and the Institute forMarine and Atmospheric Research, Utrecht (IMAU), and we wish to thankthe field party. Additional funding was provided by the Thule Institute,Nessling Foundation, Swedish Science Council, Norwegian ResearchCouncil, NARP, and Stiftelsen Ymer-80. Finally, the anonymous refereesprovided very helpful comments.

ReferencesBolzan, J. F., and V. A. Pohjola (2000), Reconstruction of the undiffusedseasonal oxygen isotope signal in central Greenland ice cores, J. Geo-phys. Res., 105(C9), 22,095–22,106.

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Cragin, J. H., A. D. Hewitt, and S. C. Colbeck (1996), Grain-scale mechan-isms influencing the elution of ions form snow, Atmos. Environ., 30,119–127.

Dansgaard, W. (1964), Stable isotopes in precipitation, Tellus, 16, 436–468.

Davies, T. D., C. E. Vincent, and P. Brimblecombe (1982), Preferentialelution of strong acids from a Norwegian ice cap, Nature, 300(5888),161–163.

Domine, F., R. Sparapani, A. Ianniello, and H. J. Beine (2004), The originof sea salt in snow on Arctic sea ice and in coastal regions, Atmos. Chem.Phys., 4, 2259–2271.

Hubbard, B., and M. Sharp (1993), Weertman regelation, multiple refreez-ing events and the isotopic evolution of the basal ice layer, J. Glaciol.,39(132), 275–291.

Iizuka, Y., M. Igarashi, K. Kamiyama, H. Motoyama, and O. Watanabe(2002), Ratios of Mg2+/Na+ in snowpack and an ice core at Austfonna icecap, Svalbard, as an indicator of seasonal melting, J. Glaciol., 48(162),452–460.

Isaksson, E., et al. (2001), A new ice core record from Lomonosovfonna,Svalbard: Viewing the data between 1920–1997 in relation to presentclimate and environmental conditions, J. Glaciol., 47(157), 335–345.

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Isaksson, E., D. Divine, J. Kohler, T. Martma, V. Pohjola, H. Motoyama,and O. Watanabe (2005a), Climate oscillations as recorded in Svalbardice core d18O records between 1200–1997 AD, Geogr. Ann., Ser. A,87(1), 203–214.

Isaksson, E., J. Kohler, J. Moore, V. Pohjola, M. Igarashi, L. Karlof,T. Martma, H. A. J. Meijer, H. Motoyama, and R. S. W. van de Wal

(2005b), Using two ice core d18O records from Svalbard to illustrateclimate and sea ice variability over the last 400 years, Holocene, 15(4),501–509.

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Kekonen, T., J. Moore, R. Mulvaney, E. Isaksson, V. Pohjola, and R. S. W.van de Wal (2002), An 800 year record of nitrate from the Lomonosov-fonna ice core, Svalbard, Ann. Glaciol., 35, 261–265.

Kekonen, T., J. Moore, P. Peramaki, R. Mulvaney, E. Isaksson, V. Pohjola,and R. S. W. van de Wal (2005), The 800 year long ion record from theLomonosovfonna (Svalbard) ice core, J. Geophys. Res., 110, D07304,doi:10.1029/2004JD005223.

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Moore, J. C., A. Grinsted, T. Kekonen, and V. Pohjola (2005), Separation ofmelting and environmental signals in an ice core with seasonal melt,Geophys. Res. Lett., 32, L10501, doi:10.1029/2005GL023039.

Nordli, P. Ø., I. Hanssen-Bauer, and E. J. Førland (1996), Homogeneityanalyses of temperature and precipitation series from Svalbard and JanMayen, DNMI-Klima Rep. 16/96, Norw. Meteorol. Inst., Oslo.

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Pohjola, V. A., J. C. Moore, E. Isaksson, T. Jauhiainen, R. S. W. van deWal, T. Martma, H. A. J. Meijer, and R. Vaikmae (2002a), Effect ofperiodic melting on geochemical and isotopic signals in an ice core fromLomonosovfonna, Svalbard, J. Geophys. Res., 107(D4), 4036,doi:10.1029/2000JD000149.

Pohjola, V. A., T. Martma, H. A. J. Meijer, J. C. Moore, E. Isaksson,R. Vaikmae, and R. S.W. van deWal (2002b), Reconstruction of 300 yearsannual accumulation rates based on the record of stable isotopes of waterfrom Lomonosovfonna, Svalbard, Ann. Glaciol., 35, 57–62.

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van de Wal, R. S. W., R. Mulvaney, E. Isaksson, J. C. Moore, J.-F. Pinglot,V. Pohjola, and M. P. A. Thomassen (2002), Reconstruction of the histor-ical temperature trend frommeasurements in amedium-length bore hole onthe Lomonosovfonna Plateau, Svalbard, Ann. Glaciol., 35, 371–378.

van Lipzig, N. P. M., E. van Meijgaard, and J. Oerlemans (2002), The effectof temporal variations in the surface mass balance and temperature-inversion strength on the interpretation of ice core signals, J. Glaciol.,48(163), 611–621.

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Virkkunen, K. (2004), Snowpit studies in 2001–2002 in Lomonosovfonna,Svalbard, M. S. thesis, 71 pp., Univ. of Oulu, Oulu, Finland.

von Storch, H., E. Zorita, J. M. Jones, Y. Dimitriev, F. Gonzalez-Rouco, andS. F. B. Tett (2004), Reconstructing past climate from noisy data, Science,306(5696), 679–682, doi:10.1126/science.1096109.

�����������������������A. Grinsted and J. C. Moore, Arctic Centre, University of Lapland, P. O.

Box 122, FIN-96101 Rovaniemi, Finland. (ag@glaciology.net)E. Isaksson, Norwegian Polar Institute, Polar Environmental Centre,

N-9296, Tromsø, Norway.T. Martma, Institute of Geology, Tallinn University of Technology,

Estonia Av. 7, 10143 Tallinn, Estonia.V. Pohjola, Department of Earth Sciences, Uppsala University, Villavagen

16, SE-752 36 Uppsala, Sweden.

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IV

1

Observational evidence for volcanic impact on sea level and the global water cycle A. Grinsted1,2, J. C. Moore1 & S. Jevrejeva3

1. Arctic Centre, University of Lapland, Rovaniemi, Finland

2. Department of Geophysics, University of Oulu, Oulu, Finland

3. Proudman Oceanographic Laboratory, Liverpool, UK

It has previously been noted that there are drops in global sea level (GSL) following some major volcanic eruptions1,2. However, observational evidence has not been convincing as there is substantial variability in the global sea level record on periods similar to those at which we expect volcanoes to have an impact3. To quantify the impact of volcanic eruptions we average monthly GSL data from 830 tide gauge records around 5 major volcanic eruptions. Surprisingly, we find that the initial response to a volcanic eruption is a significant rise in sea level of 9 mm in the first year after the eruption. This is followed by a drop of 7 mm in the period 2-3 years after the eruption relative to pre-eruption sea level. Neither the drop nor especially the rise in GSL can be explained by models of lower oceanic heat content4,5. The volcanic impact on the water cycle is comparable in magnitude to that of a large El Niño-La Niña cycle, amounting to about 5% of global land precipitation6.

Major volcanic eruptions inject scattering aerosol into the stratosphere and thus impose a significant radiative cooling of the atmosphere7. In the years following major eruptions it has been observed that there is a drop in Global Oceanic Heat Content (GOHC)5. A colder ocean is denser and an associated drop in global mean sea level of ~5mm during the 1-4 years following a major eruption has been modelled5. Stenchikov et al.8 found by modelling that large scale atmospheric circulation patterns such as the Arctic Oscillation (AO) were affected by the Pinatubo eruption. We have previously shown a link between AO and European sea level3, which is best explained by a redistribution of water driven by shifts in surface air pressure.

Producing a GSL curve with valid confidence intervals from the observational data base is not a trivial job (See Supplementary Information). We use 830 time series of monthly mean relative sea level (RSL) from the Permanent Service for Mean Sea Level (PSMSL) database9. Detailed descriptions of these time series are available from www.pol.ac.uk/psmsl. We applied an inverted barometer correction to the records using sea level pressure from the HadSLP2 dataset10, taking care to adjust the atmospheric pressure so that the over ocean pressure integral is constant. RSL data sets were corrected for local datum changes and glacial isostatic adjustment (GIA) of the solid Earth11. However, as we are interested in the transient response of GSL to volcanic eruptions we will be detrending GSL prior to analysis and therefore the choice of GIA corrections is not critical to this study. We have developed a new ‘virtual station’ method to overcome geographical bias, so that stations close to each other are weighted much less than isolated ones, and which can quantify the uncertainties due to the

2

representativeness of the stations used12. We assign each station to one of 13 regions (Fig. 1) and calculate the GSL from the average of the regional sea level rates. The reconstructed GSL will effectively be smoothed by 12 month averages due to the method used to avoid seasonal bias. The evolution and regional distribution of station numbers as well as a comparison with other GSL reconstructions1,2,12 can be found in the Supplementary Information.

There are 9 eruptions since 1890 that produced a stratospheric aerosol loading at least ~10% as big as Pinatubo in 19917 (see Supplementary information). Only eruptions located in the tropics affect both hemispheres, and the 5 largest eruptions are all tropical, with only one of the 9 located at high latitudes7. The GSL curve shows considerable variability on periods comparable to the timescale at which we expect volcanoes to have an impact, and any significant change before and after a particular volcanic eruption could be argued to be caused by natural variability such as e.g. El Niño. We therefore examine the average impact from the 5 most recent major tropical volcanic eruptions (Colima, February 1890; Santa Maria, October 1902; Agung, March 1963; El Chichón, April 1982; Pinatubo, June 1991) which have been previously modelled to have an impact on GSL5. Since El Niño-Southern Oscillation has a large impact of sea level3,12, especially in the tropics7, it is fortunate that these eruptions occurred in both El Niño and La Niña years, so that the volcanic signal is maximized at the expense of non-volcanic variability. The average detrended GSL can be seen in Fig. 2. The qualitative shape of Fig. 2 (the initial peak followed by a drop) is robust to excluding any particular volcano from the analysis. The curves for the individual eruptions, and an alternative method of calculating average impact using 9 volcanic eruptions may be seen in the Supplementary Information. The GSL response is not dominated by the signal in any single region (See Supplementary Table 4). The detrended GSL curve is far from spectrally white (the coefficient of a first order auto-regressive, AR(1)=0.98) and conventional tests overestimate the significance. For that reason, we test the significance of the main features seen in the plot by Monte Carlo sampling of the detrended GSL series using 5 random dates instead of the real eruption years. That is when testing whether the median sea level in two windows (i.e. the centre line of the boxplots in Fig. 2) are significantly different, we find the distribution of the absolute median difference from 10000 window pairs (with same length and separation). This gives a conservative estimate of the significance because the detrended GSL series presumably contains some volcanic variability.

The observed average response to the eruptions is an initial rise in GSL in the first year followed by a drop in the period 2-4 years after the eruption (Fig. 2). GSL reaches background levels ~5 years after an eruption. The maximal rise of 9 mm occurs 4 months after the eruption which coincides with the timing of the maximal radiative forcing7, and the maximal drop of 7 mm occurs after 32 months. In order to asses the significance of the rise and drop, we compare the median GSL of windows representing pre-eruption GSL (0-2 years before eruption), the rise (the first year post eruption), and the drop (2-3 years post eruption). We label these 3 windows Before, Rise, and Drop (Fig. 2). We find that the median Before-Rise difference is significant at the 99% confidence level, whereas the Before-Drop difference is only significant at the 81% level. The Drop appears less significant as it is well separated from the Before period. The Rise to Drop median difference is significant at the 99% level. The Rise and Drop

3

characteristics are robust and not dominated by any particular region or volcanic eruption.

The timing of the largest drop in GSL ~3 years after the eruption agrees with modelling and observations of GOHC4,13, however its magnitude of ~7 mm is about 40% larger than modelled5. Furthermore, we unexpectedly find a significant rise of ~9 mm in GSL in the first year immediately following an eruption. These differences between modelled and observed volcanic impact on sea level suggests that, in addition to lowering the GOHC, major volcanic eruptions affect both ocean volume and mass. Mass changes in the oceans may be mediated by variation in glacier mass balance, river discharge, and precipitation-evaporation (P-E)14.

Volcanic eruptions reduce atmospheric water vapour15. In particular, Pinatubo caused a decrease of only ~0.5 mm15 and the ~9 mm peak in GSL can therefore not be explained by drying of the atmosphere. There is both observational and modelling evidence for less precipitation over land after volcanic eruptions7,16. This reduction in precipitation has previously been tied to reduced evaporation5,7. We propose that the initial peak and the larger than expected drop in sea level, may be explained by lower evaporation from the oceans and a delayed response of river discharge to the associated lower precipitation. The volcanic radiative forcing will have an immediate cooling effect at the ocean surface. Oceanic evaporation rate is strongly determined by ocean skin temperature17, although mixing dilutes the temperature anomalies, and we expect evaporation to closely track the radiative forcing. The observed peak in sea level occurs 4 months after the eruption, which agrees well with the timing of the maximal radiative forcing after a large eruption7. In contrast, a steric drop in sea level would be caused by a reduced GOHC and would largely depend on the integrated radiative forcing and hence be delayed and of longer duration18. The observation that GSL first rises and then drops thus agrees with our hypothesis. To summarize, initially there is lower ocean evaporation but river discharge does not respond immediately to the consequent lower precipitation and therefore sea level rises. After 1 to 2 years P-E returns to normal, but river discharge is now relatively low due to the reduced land precipitation in the preceding years. Additionally the global cooling caused by the eruption reduces runoff from ice masses – effectively lowering sea level. In addition to this transient response of the water cycle there is the steric drop in GSL that may persist for many years18,5. This hypothesis is consistent with the observation (see Supplementary Information) that the initial rise in sea level is especially evident in regions where the steric change is relatively small, such as in the Baltic and Northeast Pacific19,20.

The magnitude of the impact on the water cycle may be seen by comparison with natural changes in global river discharge21 where a 5% change corresponds to a change in GSL of 5.5mm yr-1. The initial 9 mm rise in sea level is equivalent to a 2.5% change in global lake and river reservoirs22. This rise corresponds to an imbalance in ocean surface water fluxes of ~0.02 mm/day for a year, which translates into a reduction in land P-E of ~0.06 mm/day or ~3% of mean land precipitation. This is in the range of modelled natural variability23. The magnitude of this change is similar to the ~4% variations in global land precipitation associated with El Niño-Southern Oscillation6.The volcanic induced changes in water cycle may, depending on relative timing, act in

4

opposite sense to those that occur in a normal El Niño-La Niña cycle, thus supporting observations that the El Niño coincident with eruptions of El Chichón (1982/3) and Pinatubo (1991/2) were rather atypical in their oceanic precipitation/temperature behaviour6,7. The respective roles that heat content, evaporation, and inter-oceanic redistribution of water play in the regional response of sea level need further study. The proposed mechanism for the initial rise in sea level can be tested by modelling the volcanic impact on ocean evaporation, ocean precipitation, and river discharge.

<received> Style tag for received and accepted dates (omit if these are unknown).

1. Church, J. A., White, N. J., Coleman, R., Lambeck, K. & Mitrovica, J. X. Estimates of the regional distribution of sea-level rise over the 1950 to 2000 period. J. Clim. 17, 2609–-2625 (2004).

2. Church, J. A., & White N. J. A 20th century acceleration in global sea-level rise, Geophys. Res. Lett., 33, L01602, doi:10.1029.2005GL024826 (2006)

3. Jevrejeva, S., Moore, J. C., Woodworth P.L. & Grinsted A. Influence of large-scale atmospheric circulation on European sea level: results based on the wavelet transform method, Tellus 57A, 183–193 (2005).

4. Hansen, J., et al. Climate forcings in Goddard Institute for Space Studies SI2000 simulations, J. Geophys. Res., 107, 4347, doi:10.1029.2001JD001143 (2002).

5. Church, J. A., White, N. J. & Arblaster, J. M. Significant decadal-scale impact of volcanic eruptions on sea level and ocean heat content. Nature 438,doi:10.1038.nature04237 (2005)

6. Adler, B.F., et al. The Version 2 Global Precipitation Climatology Project (GPCP) Monthly Precipitation Analysis (1979-Present). J. Hydrometeorol. 4, 1147–1167, (2003).

7. Robock, A. Volcanic eruptions and climate. Rev. Geophys. 38, 191-219 (2000).

8. Stenchikov, G. et al. Arctic Oscillation response to the 1991 Mount Pinatubo eruption: Effects of volcanic aerosols and ozone depletion, J. Geophys. Res., 107, 4803, doi:10.1029.2002JD002090 (2002).

9. Woodworth, P.L. & Player R. The Permanent Service for Mean Sea Level: an update to the 21st century, J. Coastal Res., 19, 287-295 (2003).

10. Allan, R. J. & Ansell, T. J. A new globally complete monthly historical mean sea level pressure data set (HadSLP2): 1850-2004, J. Clim., accepted, (2006).

11. Peltier, W.R. Global glacial isostasy and the surface of the ice-age earth: The ICE-5G model and GRACE, Ann. Rev. Earth Planet. Sci., 32, 111-149 (2004).

12. Jevrejeva, S., A. Grinsted, J. C. Moore, and S. Holgate, Nonlinear trends and multiyear cycles in sea level records, J. Geophys. Res., 111, C09012, doi:10.1029/2005JC003229 (2006).

13. Levitus, S., Antonov, J. & Boyer T. Warming of the world ocean, 1955–2003, Geophys. Res. Lett., 32, L02604, doi:10.1029/2004GL021592 (2005).

14. Cazenave, A. & Nerem R. S. Present-day sea level change: Observations and causes, Rev. Geophys., 42, RG3001, doi:10.1029.2003RG000139 (2004).

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15. Soden, B. J., R. T. Wetherald, G. L. Stenchikov, and A. Robock: Global cooling following the eruption of Mt. Pinatubo: A test of climate feedback by water vapor. Science, 296, 727–730 (2002).

16. Gillett, N. P., Weaver, A. J., Zwiers, F. W. & Wehner M. F. Detection of volcanic influence on global precipitation, Geophys. Res. Lett., 31, L12217, doi:10.1029.2004GL020044 (2004).

17. Thomas, G. E. & Stamnes, K. Radiative transfer in the atmosphere and ocean.Cambridge Univ. Press, New York. 517 pp, Ch.1, 27-28. (1999).

18. Gleckler, P., T. Wigley, B. Santer, J. Gregory, K. AchutaRao, and K. Taylor (2006). Volcanoes and climate: Krakatoa’s signature persists in the ocean. Nature, 439,doi:10.1038/439675a. (2006)

19. Antonov, J.I., Levitus, S. & Boyer T. P. Thermosteric sea level rise, 1955–2003 Geophys. Res. Lett 32, L12602, doi:10.1029.2005GL023112 (2005).

20. Cabanes, C., Cazenave, A. & Le Provost C. Sea Level Rise During Past 40 Years Determined from Satellite and in Situ Observations, Science, 294(5543), 840-842 (2001).

21. Fekete, B. M., Vörösmarty, C. J. & Grabs W. Global, Composite Runoff Fields Based on Observed River Discharge and Simulated Water Balances (Report No. 22, GRDC, Koblenz, Germany, 2nd ed.) (1999).

22. Gleick, P. H.: Water resources. In Encyclopedia of Climate and Weather, ed. by S. H. Schneider, Oxford University Press, New York, vol. 2, pp.817-823 (1996).

23. Broccoli, A. J., et al. Twentieth-century temperature and precipitation trends in ensemble climate simulations including natural and anthropogenic forcing, J. Geophys. Res., 108, 4798, doi:10.1029.2003JD003812 (2003). Supplementary Information is linked to the online version of the paper at www.nature.com/nature

Acknowledgements Funding was provided by the Thule institute, the Maj and Tor Nessling foundation, and the Finnish Academy.

Author Information Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to A.G. (ag@glaciology.net).

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Antarctic

Arctic

Baltic

CPacific

Indian

Mediterr

NEAtlantic

NEPacific

NWAtlantic

SEPacific

SEatlantic

SWatlantic

WPacific

Longitude

Latit

ude

0 90 180 270 360−90

−45

0

45

90

Figure 1. Location of tide gauges and their distribution into 13 regions. Symbols indicate the designated region.

Before

Rise

Drop

Year relative to eruption

Det

rend

ed G

SL

(mm

)

−6 −4 −2 0 2 4 6−15

−10

−5

0

5

10

15

20

Figure 2. Average impact of 5 major volcanoes on sea level. The solid black curve shows non-linearly detrended (See Ref. 12 and Supplementary Information) GSL as observed in tide gauge records (grey shows range of values). Boxplots show the median sea level (centre line of box) of the “Before”, “Rise” and “Drop” windows which we compare in the text. Notches show the 95% confidence interval of the median, top and bottom edges of each box are the 25th and 75th percentiles and the full range can be seen from the grey shaded area.

1

Observational evidence for volcanic impact on sea level and the global water cycle A. Grinsted1,2, J. C. Moore1 & S. Jevrejeva3

1. Arctic Centre, University of Lapland, Rovaniemi, Finland

2. Department of Geophysics, University of Oulu, Oulu, Finland

3. Proudman Oceanographic Laboratory, Liverpool, UK

It has previously been noted that there are drops in global sea level (GSL) following some major volcanic eruptions1,2. However, observational evidence has not been convincing as there is substantial variability in the global sea level record on periods similar to those at which we expect volcanoes to have an impact3. To quantify the impact of volcanic eruptions we average monthly GSL data from 830 tide gauge records around 5 major volcanic eruptions. Surprisingly, we find that the initial response to a volcanic eruption is a significant rise in sea level of 9 mm in the first year after the eruption. This is followed by a drop of 7 mm in the period 2-3 years after the eruption relative to pre-eruption sea level. Neither the drop nor especially the rise in GSL can be explained by models of lower oceanic heat content4,5. The volcanic impact on the water cycle is comparable in magnitude to that of a large El Niño-La Niña cycle, amounting to about 5% of global land precipitation6.

Major volcanic eruptions inject scattering aerosol into the stratosphere and thus impose a significant radiative cooling of the atmosphere7. In the years following major eruptions it has been observed that there is a drop in Global Oceanic Heat Content (GOHC)5. A colder ocean is denser and an associated drop in global mean sea level of ~5mm during the 1-4 years following a major eruption has been modelled5. Stenchikov et al.8 found by modelling that large scale atmospheric circulation patterns such as the Arctic Oscillation (AO) were affected by the Pinatubo eruption. We have previously shown a link between AO and European sea level3, which is best explained by a redistribution of water driven by shifts in surface air pressure.

Producing a GSL curve with valid confidence intervals from the observational data base is not a trivial job (See Supplementary Information). We use 830 time series of monthly mean relative sea level (RSL) from the Permanent Service for Mean Sea Level (PSMSL) database9. Detailed descriptions of these time series are available from www.pol.ac.uk/psmsl. We applied an inverted barometer correction to the records using sea level pressure from the HadSLP2 dataset10, taking care to adjust the atmospheric pressure so that the over ocean pressure integral is constant. RSL data sets were corrected for local datum changes and glacial isostatic adjustment (GIA) of the solid Earth11. However, as we are interested in the transient response of GSL to volcanic eruptions we will be detrending GSL prior to analysis and therefore the choice of GIA corrections is not critical to this study. We have developed a new ‘virtual station’ method to overcome geographical bias, so that stations close to each other are weighted much less than isolated ones, and which can quantify the uncertainties due to the

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representativeness of the stations used12. We assign each station to one of 13 regions (Fig. 1) and calculate the GSL from the average of the regional sea level rates. The reconstructed GSL will effectively be smoothed by 12 month averages due to the method used to avoid seasonal bias. The evolution and regional distribution of station numbers as well as a comparison with other GSL reconstructions1,2,12 can be found in the Supplementary Information.

There are 9 eruptions since 1890 that produced a stratospheric aerosol loading at least ~10% as big as Pinatubo in 19917 (see Supplementary information). Only eruptions located in the tropics affect both hemispheres, and the 5 largest eruptions are all tropical, with only one of the 9 located at high latitudes7. The GSL curve shows considerable variability on periods comparable to the timescale at which we expect volcanoes to have an impact, and any significant change before and after a particular volcanic eruption could be argued to be caused by natural variability such as e.g. El Niño. We therefore examine the average impact from the 5 most recent major tropical volcanic eruptions (Colima, February 1890; Santa Maria, October 1902; Agung, March 1963; El Chichón, April 1982; Pinatubo, June 1991) which have been previously modelled to have an impact on GSL5. Since El Niño-Southern Oscillation has a large impact of sea level3,12, especially in the tropics7, it is fortunate that these eruptions occurred in both El Niño and La Niña years, so that the volcanic signal is maximized at the expense of non-volcanic variability. The average detrended GSL can be seen in Fig. 2. The qualitative shape of Fig. 2 (the initial peak followed by a drop) is robust to excluding any particular volcano from the analysis. The curves for the individual eruptions, and an alternative method of calculating average impact using 9 volcanic eruptions may be seen in the Supplementary Information. The GSL response is not dominated by the signal in any single region (See Supplementary Table 4). The detrended GSL curve is far from spectrally white (the coefficient of a first order auto-regressive, AR(1)=0.98) and conventional tests overestimate the significance. For that reason, we test the significance of the main features seen in the plot by Monte Carlo sampling of the detrended GSL series using 5 random dates instead of the real eruption years. That is when testing whether the median sea level in two windows (i.e. the centre line of the boxplots in Fig. 2) are significantly different, we find the distribution of the absolute median difference from 10000 window pairs (with same length and separation). This gives a conservative estimate of the significance because the detrended GSL series presumably contains some volcanic variability.

The observed average response to the eruptions is an initial rise in GSL in the first year followed by a drop in the period 2-4 years after the eruption (Fig. 2). GSL reaches background levels ~5 years after an eruption. The maximal rise of 9 mm occurs 4 months after the eruption which coincides with the timing of the maximal radiative forcing7, and the maximal drop of 7 mm occurs after 32 months. In order to asses the significance of the rise and drop, we compare the median GSL of windows representing pre-eruption GSL (0-2 years before eruption), the rise (the first year post eruption), and the drop (2-3 years post eruption). We label these 3 windows Before, Rise, and Drop (Fig. 2). We find that the median Before-Rise difference is significant at the 99% confidence level, whereas the Before-Drop difference is only significant at the 81% level. The Drop appears less significant as it is well separated from the Before period. The Rise to Drop median difference is significant at the 99% level. The Rise and Drop

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characteristics are robust and not dominated by any particular region or volcanic eruption.

The timing of the largest drop in GSL ~3 years after the eruption agrees with modelling and observations of GOHC4,13, however its magnitude of ~7 mm is about 40% larger than modelled5. Furthermore, we unexpectedly find a significant rise of ~9 mm in GSL in the first year immediately following an eruption. These differences between modelled and observed volcanic impact on sea level suggests that, in addition to lowering the GOHC, major volcanic eruptions affect both ocean volume and mass. Mass changes in the oceans may be mediated by variation in glacier mass balance, river discharge, and precipitation-evaporation (P-E)14.

Volcanic eruptions reduce atmospheric water vapour15. In particular, Pinatubo caused a decrease of only ~0.5 mm15 and the ~9 mm peak in GSL can therefore not be explained by drying of the atmosphere. There is both observational and modelling evidence for less precipitation over land after volcanic eruptions7,16. This reduction in precipitation has previously been tied to reduced evaporation5,7. We propose that the initial peak and the larger than expected drop in sea level, may be explained by lower evaporation from the oceans and a delayed response of river discharge to the associated lower precipitation. The volcanic radiative forcing will have an immediate cooling effect at the ocean surface. Oceanic evaporation rate is strongly determined by ocean skin temperature17, although mixing dilutes the temperature anomalies, and we expect evaporation to closely track the radiative forcing. The observed peak in sea level occurs 4 months after the eruption, which agrees well with the timing of the maximal radiative forcing after a large eruption7. In contrast, a steric drop in sea level would be caused by a reduced GOHC and would largely depend on the integrated radiative forcing and hence be delayed and of longer duration18. The observation that GSL first rises and then drops thus agrees with our hypothesis. To summarize, initially there is lower ocean evaporation but river discharge does not respond immediately to the consequent lower precipitation and therefore sea level rises. After 1 to 2 years P-E returns to normal, but river discharge is now relatively low due to the reduced land precipitation in the preceding years. Additionally the global cooling caused by the eruption reduces runoff from ice masses – effectively lowering sea level. In addition to this transient response of the water cycle there is the steric drop in GSL that may persist for many years18,5. This hypothesis is consistent with the observation (see Supplementary Information) that the initial rise in sea level is especially evident in regions where the steric change is relatively small, such as in the Baltic and Northeast Pacific19,20.

The magnitude of the impact on the water cycle may be seen by comparison with natural changes in global river discharge21 where a 5% change corresponds to a change in GSL of 5.5mm yr-1. The initial 9 mm rise in sea level is equivalent to a 2.5% change in global lake and river reservoirs22. This rise corresponds to an imbalance in ocean surface water fluxes of ~0.02 mm/day for a year, which translates into a reduction in land P-E of ~0.06 mm/day or ~3% of mean land precipitation. This is in the range of modelled natural variability23. The magnitude of this change is similar to the ~4% variations in global land precipitation associated with El Niño-Southern Oscillation6.The volcanic induced changes in water cycle may, depending on relative timing, act in

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opposite sense to those that occur in a normal El Niño-La Niña cycle, thus supporting observations that the El Niño coincident with eruptions of El Chichón (1982/3) and Pinatubo (1991/2) were rather atypical in their oceanic precipitation/temperature behaviour6,7. The respective roles that heat content, evaporation, and inter-oceanic redistribution of water play in the regional response of sea level need further study. The proposed mechanism for the initial rise in sea level can be tested by modelling the volcanic impact on ocean evaporation, ocean precipitation, and river discharge.

<received> Style tag for received and accepted dates (omit if these are unknown).

1. Church, J. A., White, N. J., Coleman, R., Lambeck, K. & Mitrovica, J. X. Estimates of the regional distribution of sea-level rise over the 1950 to 2000 period. J. Clim. 17, 2609–-2625 (2004).

2. Church, J. A., & White N. J. A 20th century acceleration in global sea-level rise, Geophys. Res. Lett., 33, L01602, doi:10.1029.2005GL024826 (2006)

3. Jevrejeva, S., Moore, J. C., Woodworth P.L. & Grinsted A. Influence of large-scale atmospheric circulation on European sea level: results based on the wavelet transform method, Tellus 57A, 183–193 (2005).

4. Hansen, J., et al. Climate forcings in Goddard Institute for Space Studies SI2000 simulations, J. Geophys. Res., 107, 4347, doi:10.1029.2001JD001143 (2002).

5. Church, J. A., White, N. J. & Arblaster, J. M. Significant decadal-scale impact of volcanic eruptions on sea level and ocean heat content. Nature 438,doi:10.1038.nature04237 (2005)

6. Adler, B.F., et al. The Version 2 Global Precipitation Climatology Project (GPCP) Monthly Precipitation Analysis (1979-Present). J. Hydrometeorol. 4, 1147–1167, (2003).

7. Robock, A. Volcanic eruptions and climate. Rev. Geophys. 38, 191-219 (2000).

8. Stenchikov, G. et al. Arctic Oscillation response to the 1991 Mount Pinatubo eruption: Effects of volcanic aerosols and ozone depletion, J. Geophys. Res., 107, 4803, doi:10.1029.2002JD002090 (2002).

9. Woodworth, P.L. & Player R. The Permanent Service for Mean Sea Level: an update to the 21st century, J. Coastal Res., 19, 287-295 (2003).

10. Allan, R. J. & Ansell, T. J. A new globally complete monthly historical mean sea level pressure data set (HadSLP2): 1850-2004, J. Clim., accepted, (2006).

11. Peltier, W.R. Global glacial isostasy and the surface of the ice-age earth: The ICE-5G model and GRACE, Ann. Rev. Earth Planet. Sci., 32, 111-149 (2004).

12. Jevrejeva, S., A. Grinsted, J. C. Moore, and S. Holgate, Nonlinear trends and multiyear cycles in sea level records, J. Geophys. Res., 111, C09012, doi:10.1029/2005JC003229 (2006).

13. Levitus, S., Antonov, J. & Boyer T. Warming of the world ocean, 1955–2003, Geophys. Res. Lett., 32, L02604, doi:10.1029/2004GL021592 (2005).

14. Cazenave, A. & Nerem R. S. Present-day sea level change: Observations and causes, Rev. Geophys., 42, RG3001, doi:10.1029.2003RG000139 (2004).

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15. Soden, B. J., R. T. Wetherald, G. L. Stenchikov, and A. Robock: Global cooling following the eruption of Mt. Pinatubo: A test of climate feedback by water vapor. Science, 296, 727–730 (2002).

16. Gillett, N. P., Weaver, A. J., Zwiers, F. W. & Wehner M. F. Detection of volcanic influence on global precipitation, Geophys. Res. Lett., 31, L12217, doi:10.1029.2004GL020044 (2004).

17. Thomas, G. E. & Stamnes, K. Radiative transfer in the atmosphere and ocean.Cambridge Univ. Press, New York. 517 pp, Ch.1, 27-28. (1999).

18. Gleckler, P., T. Wigley, B. Santer, J. Gregory, K. AchutaRao, and K. Taylor (2006). Volcanoes and climate: Krakatoa’s signature persists in the ocean. Nature, 439,doi:10.1038/439675a. (2006)

19. Antonov, J.I., Levitus, S. & Boyer T. P. Thermosteric sea level rise, 1955–2003 Geophys. Res. Lett 32, L12602, doi:10.1029.2005GL023112 (2005).

20. Cabanes, C., Cazenave, A. & Le Provost C. Sea Level Rise During Past 40 Years Determined from Satellite and in Situ Observations, Science, 294(5543), 840-842 (2001).

21. Fekete, B. M., Vörösmarty, C. J. & Grabs W. Global, Composite Runoff Fields Based on Observed River Discharge and Simulated Water Balances (Report No. 22, GRDC, Koblenz, Germany, 2nd ed.) (1999).

22. Gleick, P. H.: Water resources. In Encyclopedia of Climate and Weather, ed. by S. H. Schneider, Oxford University Press, New York, vol. 2, pp.817-823 (1996).

23. Broccoli, A. J., et al. Twentieth-century temperature and precipitation trends in ensemble climate simulations including natural and anthropogenic forcing, J. Geophys. Res., 108, 4798, doi:10.1029.2003JD003812 (2003). Supplementary Information is linked to the online version of the paper at www.nature.com/nature

Acknowledgements Funding was provided by the Thule institute, the Maj and Tor Nessling foundation, and the Finnish Academy.

Author Information Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to A.G. (ag@glaciology.net).

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Antarctic

Arctic

Baltic

CPacific

Indian

Mediterr

NEAtlantic

NEPacific

NWAtlantic

SEPacific

SEatlantic

SWatlantic

WPacific

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ude

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Figure 1. Location of tide gauges and their distribution into 13 regions. Symbols indicate the designated region.

Before

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SL

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−6 −4 −2 0 2 4 6−15

−10

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Figure 2. Average impact of 5 major volcanoes on sea level. The solid black curve shows non-linearly detrended (See Ref. 12 and Supplementary Information) GSL as observed in tide gauge records (grey shows range of values). Boxplots show the median sea level (centre line of box) of the “Before”, “Rise” and “Drop” windows which we compare in the text. Notches show the 95% confidence interval of the median, top and bottom edges of each box are the 25th and 75th percentiles and the full range can be seen from the grey shaded area.

V

Instruments and Methods

Singular spectrum analysis and envelope detection:methods of enhancing the utility of ground-penetrating

radar data

John C. MOORE, Aslak GRINSTEDArctic Centre, University of Lapland, Box 122, FIN-96101 Rovaniemi, Finland

E-mail: jmoore@ulapland.fi

ABSTRACT. We present a novel method of improving signal-to-noise ratio in radargrams. The methoduses singular spectrum analysis (SSA) to separate each individual radar trace into orthogonalcomponents. The components that explain most of the original trace variance contain mainly physicallymeaningful signal, while the components with little variance tend to be noise. Adding the largest-magnitude components together until the sum of components accounts for the variance above the noiselevel (typically 60–80%%) of the original trace variance results in a much cleaner radargram with moreeasily seen internal features than in traditionally filtered data. The radargrams can be further enhancedby envelope-detecting the SSA-filtered data, as this measure of instantaneous energy minimizes thedeleterious effects of innumerable phase changes at dielectric boundaries. Subsequent incoherentstacking results in far more structured radargrams than are achieved with traditionally processed radardata and amplitude stacking.

INTRODUCTIONGround-penetrating radar (GPR) is a very widely usedglaciological tool. Commonly GPRs are impulse radars thatoperate between a few MHz and 1GHz (e.g. Sinisalo andothers, 2003). Lower frequencies give greater penetrationbut lower resolution, while higher frequencies gain reso-lution at the expense of much less penetration (e.g. Wattsand England, 1976). Frequently it is not simply the depth tobed that is of primary interest, but the internal reflectionspresent in the ice (e.g. Miners and others, 1998; Eisen andothers, 2003; Sinisalo and others, 2003). These reflectionsare usually much weaker than the bed reflector, and usuallyrequire more processing power and effort to recover. Thevast amount of data usually collected during surveys isseldom exploited beyond the standard output available fromfairly elementary qualitative processing and interpretationsoftware such as Radan (Geophysical Survey Systems Inc.)or Haescan (Roadscanners Oy). In glaciological researchthere are frequently prohibitive logistical difficulties inreturning to a particular glacier to collect additional datathat may aid interpretation. Therefore it is worth going tomore trouble than would be the case in a commercial surveyto extract the maximum glaciological information fromexisting radar data. Typically, radar data processed withcommercial packages, which are aimed more at thecommercial user than at the researcher, are often manipu-lated in ways that are not easy to understand as the softwaresource code is private. This severely limits the additionalprocessing researchers can do on the output from thecommercial software.

Here we present a simple method that is easilyprogrammed (e.g. in Matlab2), and which can be used ondata in a complementary way to commercial software,either on raw data or on output that has been manipulatedby other software.

METHODSSingular spectrum analysis (SSA) was developed as a meansof analyzing time series (Allen and Smith, 1996; Ghil andothers, 2002). The method is particularly suited to short,noisy time series such as those often found in palaeoclimatedatasets. An individual radar trace spans the moment theradar transmitter antenna is triggered to the end of the timeperiod that contains useful reflections. These radar traces aretypically only about 1 ms in length, yet they can be analyzedby any time-series method. The method we describe is not aspectral technique; it operates in the time domain, and assuch is suited to time series containing quasi-periodicsignals, rather than strict sinusoids. This is the situation witha radar trace produced by reflections of the transmittedwavelet from the virtually infinite number of subtle dielectricboundaries in a real glacier. The recorded trace contains acontinuously varying power spectrum as the radar waveformis modified over its propagation path through the glacier bythe constantly changing ice dielectric properties. This is thecase in polar ice, despite density contrasts being rathersmooth, as the dielectric constant of firn changes by a factortwo in the upper 100m of the ice sheet due to densityincrease alone. Thus, accurate modelling of radar traces inAntarctica requires a time-domain finite-difference solution(e.g. Miners and other, 1998; Eisen and others, 2003). Intemperate or polythermal glaciers, where melt layers causerapid changes in density and dielectric properties, the radarwavelet changes even more dramatically with depth. Thismeans that simple frequency filtering is unlikely to be a goodway of increasing the signal-to-noise ratio, and in ourexperience it seldom clarifies radargrams.

We decompose each of the radar trace time series usingSSA (Allen and Smith, 1996) with a set of data-adaptive,orthogonal filters. SSA amplifies signal-to-noise ratio byseparating the original time series into low-frequency trends

Journal of Glaciology, Vol. 52, No. 176, 2006 159

and narrowband quasi-periodic signals, with the rest(assumed to be noise) distributed among the filters. Thenumber of orthogonal filters used is called the embeddingdimension and is an important parameter in SSA. SSAoperates by examining the dynamics of a complex systemusing the time-lagging method (e.g. Abarbanel, 1996). Inthis method the only information on the dynamics of thesystem comes from a time series of observations, in this casethe voltages on the receiver antenna. A phase space that willapproximate the dynamics of the system is constructed fromthe coordinates of the original time series and delayedcopies of it. Each copy is delayed by a different integermultiple of time known as the lag. The lag is often, as it ishere, the time interval between successive samples in thetime series. The total number of time-series copies definesthe embedding dimension. The smaller the embeddingspace, the shorter the length of the window over which theresolved components are calculated, and the less resolved iseach component. The longer the window, the greater thefrequency resolution of each component, but the greater thechance that noise is mistaken for signal and a greaterproportion of the time series is affected by the databoundaries. The embedding dimension also determinesthe low-frequency limit below which oscillations willappear as trends (Moore and others, 2005). In typicaltime-series analysis we would calculate the statisticalsignificance of individual components by testing thevariance explained by the component against suitable noisemodels using Monte Carlo methods. The Monte Carlomethods are much the slowest computational part of theprocedure; however, in the case of radar data we do notneed to do this as we have many time series availableand the significance of signals can be directly assessed bythe observer.

The set of incrementally time-lagged radar traces make upthe lagged-data matrix, representing the dynamical phase

space. The basic idea behind SSA is to linearly transform thehighly correlated set of vectors in the lagged-data matrix intoa set of vectors that are linearly independent. The depend-ence of the lagged data can be expressed in terms of thecovariance matrix (C ) of the lagged-data matrix. Theeigenvectors ðpk , k ¼ 1 . . .MÞ of C are the principaldirections in the data matrix and are often called empiricalorthogonal functions (EOFs), where M is the embeddingdimension. Principal components (PCs) are obtained byprojecting the lagged-data matrix onto the eigenvectors. Theeigenvalues (Vk) of C tell us how much variance is capturedby each PC and the eigenvectors and eigenvalues are sortedby importance so that V1 is largest. The PCs that capture littleof the variance (low eigenvalues, Vk) can be considerednoise. The original time series can be reconstructed from thefull set of PCs and EOFs. This can be done so that thereconstructed series is a sum of a set of reconstructedcomponents (RCs) where each RC is only derived from onePC and its associated EOF. Hence, it is possible to make apartial reconstruction of the time series where componentsthat capture little variance and are thus thought to be noisecan be excluded. We can write the reconstructed com-ponents as

RkðtÞ ¼ �kðtÞ � �kð�tÞ � XðtÞM

,

where * indicates convolution, t is time and X is the timeseries. For radar filtering, the eigenvectors can be derivedfrom the lag covariance matrix either of the full radargram oron a trace-by-trace basis. Figure 1 shows the RCs found for asingle radar trace, and the associated eigenvectors derivedfrom the covariance matrix of 2000 radar traces.

A wise choice of embedding dimension can be madewith a priori insight. However, our experience with radardata suggests this choice is not particularly critical as weare mainly interested in the noise reduction properties ofSSA as opposed to examining the particular SSA RCs thatmay correspond with physically meaningful oscillationsor trends in the radar data. As larger embeddingdimensions carry a computational penalty and are af-fected more by the data boundaries, we use rather smallvalues of M: typically a value of 20 or 30 is sufficient toperform adequately. We recommend that a wide range ofM is experimented with to examine possible unwantedartefacts.

For radar data, we simply sum the reconstructed com-ponents that account for the fraction of the original time-series variance that is above the noise floor of the data. Thenoise floor can be found by plotting the magnitude of Vk asa function of their index k when ordered by magnitude(Fig. 1), or equivalently by considering the contribution tothe variance of the original radar trace of the RCs. Oftenthere will be rapid fall-off of Vk with k until a fairly stablelevel is reached. This stable level represents the noise floor,and the components at this level are then representingmainly noise. The variance accounted for by the numbercomponents above the noise floor is then used as thenumber of RCs to be used in the SSA filtering. Typically, inour experience, the useful signal is 60–80% of the totalvariance in the radar trace. Figure 1 exhibits two rapiddrops in variance (after the first two components and afterthe third and fourth components), then there is a fairlygradual drop-off in variance accounted for by the RCs ratherthan a low-level plateau. This example presents a more

Fig. 1. SSA reconstructed components found by projecting the SSAfilters found using the whole 2000 traces in Figure 4, on tracenumber 1, ordered by magnitude of variance accounted for in theradar trace. The noise floor we estimate as being after the first sixcomponents, which accounts for 63% of variance.

Moore and Grinsted: Instruments and methods160

difficult situation than arises when the noise floor is veryclear, but there is still only a slow drop-off in varianceaccounted for beyond the sixth component. Here, wesomewhat arbitrarily select six components as the cut-off ofsignal from noise, equivalent to accounting for a varianceof 63%.

Removal of the non-linear trend component in SSA (e.g.Moore and others, 2005) corresponds with the ‘d.c. removal’or ‘de-wowing’ processes usually done in radar processing,whereby the shift in reference voltage level and very long-period oscillations in each radar trace are removed. The SSAapproach has the advantage of being a purely objectivemethod of selecting parts of the received antenna voltagepower spectrum that correspond with meaningful signals. Acommon method of attempting a similar process is to usebandpass filtering. However, particular SSA filters oftencorrespond with rather narrowband quasi-period oscilla-tions, and they can also come from anywhere in the full-frequency spectrum of the received radar trace rather thanrepresenting high- and low-pass limits. The differencebetween SSA and bandpass filtering can be seen bycomparing the power spectrum of the amplified radarreceived data with the power spectrum of the SSA com-ponents that together comprise the reconstructed signal(Fig. 2).

ENVELOPE DETECTION STACKINGOne common method of improving signal-to-noise ratio inradar is stacking. This method relies on coherent signalsadding, while noise, being incoherent, rises only as thesquare root of the stacking number. This should occur whereradar data are collected with the antennas stationary.Usually, however, radar data are collected while theantennas are continuously being moved over the snowsurface. Thus stacking is done by averaging the radar tracescollected over an appreciable distance of horizontal travel.When the radar frequencies used are high, and the dielectricproperties of the firn and ice vary rapidly laterally, theassumption of coherence may fail. Since the bandwidth of

impulse GPR is high, changes in dielectric layers, even of afew centimetres over distances between radar trace collec-tion, (commonly some decimetres) can cause undesirablephase mismatching in stacking. In our experience, this oftenoccurs, and stacking usually fails to bring significantimprovement in signal-to-noise ratio.

Signal envelope detection (Nye and Berry, 1974) is away of processing the data such that they can be stacked inan incoherent way, but which does significantly improvethe signal-to-noise ratio. This method essentially squaresand sums the real measured antenna voltage, and itsquadrature, that is equivalent to the magnitude of theHilbert transformation of the radar trace. In practice, thiscan be simply and very efficiently implemented bysumming the squared radar trace data and their squarednumerical differences, since the differential or integral of asinusoid is just a phase shift of �/2. In some commercialprocessing packages, the magnitude of the Hilbert trans-formation is directly implemented. The envelope calculatedrepresents the energy around the particular data point inthe radar trace, so the envelope is not affected by the phasechanges caused by the many dielectric discontinuities inthe firn. Therefore small-scale lateral inhomogeneity thatcan undermine coherent stacking does not adversely affectthe envelope-detected stack, and hence its usefulness forGPR data.

PRACTICAL EXAMPLES OF THE METHODFigure 3 shows all the steps in processing a radar trace thatwe use in the example radargrams discussed next.

To illustrate the advantages of the method we describeover normal processing, we show some radar data collectedin Antarctica during December 1999 (Sinisalo and others,2003). The first example is of a radar survey on Plogbreen,

Fig. 2. The normalized power spectral density found using themaximum entropy method of order 20, for a block of 50 traces ofradar data used to make one stacked trace in Figure 5. Note thepeak in power at around 2.6 cycles ns–1 (2600MHz) that is kept assignal by SSA that would very likely be rejected by bandpass-filtering the 800MHz centre frequency radar data.

Fig. 3. The steps in the processing of radar trace No. 1000 inFigure 4. (a) The originally recorded radar trace; (b) after removal ofthe mean background trace; (c) simple linear gaining; (d) afterremoval of the noise SSA components; and (e) after envelopedetection. Signal scaling is arbitrary.

Moore and Grinsted: Instruments and methods 161

about 6 km from the Finnish research station Aboa (738030 S,138250 W). The radar used was a Mala Geoscience Ramacsystem with an 800MHz centre frequency shielded-antennasystem. The radar control unit, computer and a roving globalpositioning system (GPS) receiver were mounted on thesnowmobile. The antenna system was pulled about 2mbehind the snowmobile, using rigid struts to keep it at a fixeddistance. No trace stacking was done and data werecollected on a laptop computer. Each trace contains 1024samples spanning a time window of 197 ns. Traces werecollected at 0.25 s intervals, which correspond to horizontaldistances of about 0.8m at the snowmobile travelling speed.We reproduced, using Matlab2, the processing done bySinisalo and others (2003). Background noise was removedand vertical bandpass filtering was performed. Figure 4shows an example containing 2000 traces from the radarline, a distance of about 1600m.

In our new method we do the same initial processingsteps on each trace (see Fig. 3), but instead of the frequencyfiltering we perform SSA. As discussed above, we select anembedding dimension that is relatively short; here we use avalue of 20. Figures 1 and 3 show the results of applying SSAto a single trace. The larger-magnitude reconstructed com-ponents in Figure 1 have clear changes in amplitude, as maybe expected at an internal reflection horizon, but many ofthe lesser-magnitude components also possess that feature.We think that much more could be achieved by analyzingthese components in more detail, as has been done with SSAquasi-periodic components in palaeoclimate time series,which reveal rich structure in the underlying climate system(e.g. Jevrejeva and others, 2004).

Figure 1 suggests that about 60–70% of the variance isabove the noise floor and that the six largest reconstructed

components amount to 63% of the original trace variance.We therefore extract the six leading RCs from all the 2000traces in the section shown in Figure 4, to give the SSA-processed image in Figure 4b. There is clearly an enhance-ment in the SSA-processed image in Figure 4b relative to thenormally processed data of Figure 4a. It is clear that theradar data contain eight to ten gently convex-up internalreflection bands, with hints of more detailed structure inplaces. In particular, there are reflecting layers in the deeperhalf-profile visible between traces 1500 and 2000 that arecompletely lost in the normally processed data. The layeringis even more obvious in the envelope-detected image inFigure 4c.

It is of interest to consider the cause of the layer re-flections. Sinisalo and others (2003) show that accumulationrates in the area are about 300mmw.e. a–1, and this wouldbe about 500mm of snow and firn in the upper layers.Given a radar wave velocity of 200m ms–1, 500mm thicklayers are about 5 ns of two-way travel time. Thus thelayering is not simple annual layers, but more likelyvariation in densities due to the occasional melt events inthe area.

The second example we consider examines the advan-tages of stacking GPR data. Again we consider 800MHzRamac radar data collected using the same set-up asdescribed earlier. Each trace consisted of 1024 samples thatwere collected in a time window of 148 ns, with tracescollected at 0.1 s intervals, which corresponds with threetraces per metre. The data come from an area (748340 S,108430 W) on Ritscherflya, about 15 km from the SwedishSvea station and near to the site of a Dutch automaticweather station. In total, 23 384 traces were collected, andwe process the data as described earlier, but with 50 timesstacking. SSA processing showed that about 80% of datawere above the noise floor. The reconstructed componentsaccounting for 80% of variance were summed andenvelope-filtered and then stacked 50 times. Figure 5 showsthe results: no layering can be seen in the normallyprocessed and stacked radargram (Fig. 5a), while layers arequite easily seen in the SSA- and envelope-processed data(Fig. 5b). In this example, the envelope detection part of theprocessing is responsible for most of the improvement,

Fig. 4. (a) A profile of 800MHz data from Antarctica (Sinisalo andothers, 2003), that has had the mean background trace removedand linear gaining applied. (b, c) The same image after SSA-processing (b) and after envelope-detecting (c) the radar traces.

Fig. 5. (a) Radar data processed by background removal, d.c.removal, gaining, de-wowing and high-pass filtering, and stacking50 times. (b) The same data processed by background removal,gaining, SSA filtering, envelope-detecting and stacking 50 times.

Moore and Grinsted: Instruments and methods162

mainly because there was relatively little noise present in theradar data for SSA filtering to remove. Stacking was neededin this example to reveal layering because of the very smalldensity differences present in the snow at this site. Noseasonal melting occurs and wind speeds are high, ensuringconsiderable surface sastrugi and disturbance to annuallayering. The layers are about 5 ns apart in Figure 5,suggesting that they, in contrast with the layers in Figure 4,probably represent annual stratification.

CONCLUSIONSWe present a novel method of processing radar data thatshould be especially well suited to impulse GPR systems.The SSA method is superior to common bandpass filteringapproaches to improving signal-to-noise ratio in severalways: (1) an objective method of separating signal and noisecomponents of the radar trace comes from analysis of thevariance accounted for by the orthogonal reconstructedcomponents; (2) SSA non-linear trend terms correspond withthe separate ‘d.c. level’ and ‘de-wowing’ correctionscommonly done to radar data; (3) the full power spectrumof the radar trace is available for analysis rather thanarbitrarily setting bandpass filtering limits. The individualSSA reconstructed components also merit examinationseparately as they contain information on aspects of thecomplex system comprising the electromagnetic waveinteraction with the dielectric medium of sub-parallel firnand ice layers.

We show that the GPR traces that have been ‘envelope-detected’ can be stacked much more successfully than canbe done simply by amplitude stacking as a means ofimproving signal-to-noise ratio. This is because in high-bandwidth GPR data small lateral changes in the dielectricproperties of firn layers cause phase shifts large enough todestroy coherent amplitude stacking in many practicalsurveys. Envelope detection is also unaffected by electro-magnetic phase changes on reflection, as it is a measureof the instantaneous energy at particular times in theradar trace.

ACKNOWLEDGEMENTSThe field logistics in 1999–2002 were provided by theFinnish Antarctic Research Programme (FINNARP) 1999–2002, and we thank A. Sinisalo for help with radar datacollection. The work was funded by the Finnish Academyand the Thule Institute. B. Hubbard and R. Petterssonprovided helpful comments on the manuscript.

REFERENCESAbarbanel, H.D.I. 1996. Analysis of observed chaotic data. New

York, Springer.Allen, M.R. and L.A. Smith. 1996. Monte Carlo SSA: detecting

irregular oscillations in the presence of colored noise. J. Climate,9(12), 3373–3404.

Eisen, O., F. Wilhelms, U. Nixdorf and H. Miller. 2003. Revealingthe nature of radar reflections in ice: DEP-based FDTD forwardmodeling. Geophys. Res. Lett., 30(5), 1218–1221.

Ghil, M. and 10 others. 2002. Advanced spectral methods forclimatic time series. Rev. Geophys., 40(1), 1003. (10.1029/2000RG000092.)

Jevrejeva, S., J.C. Moore and A. Grinsted. 2004. Oceanic andatmospheric transport of multiyear El Nino–Southern Oscillation(ENSO) signatures to the polar regions. Geophys. Res. Lett.,31(24), L24210. (10.1029/2004GL020871.)

Miners, W.D., N. Blindow and S. Gerland. 1998. Modelling a GPRrecord by using ice core data to produce a synthetic radiogram.In Proceedings of the Seventh International Conference onGround-Penetrating Radar, May 27–30 1998, Lawrence, Kansas,USA. Lawrence, KS, Radar Systems and Remote SensingLaboratory, 437–442.

Moore, J.C., A. Grinsted and S. Jevrejeva. 2005. New tools foranalyzing time series relationships and trends. EOS Trans. AGU,86(24), 226, 232.

Nye, J.F. and M. Berry. 1974. Dislocations in wave trains. Proc. R.Soc. London A, 336, 165–190.

Sinisalo, A., A. Grinsted, J.C. Moore, E. Karkas and R. Pettersson.2003. Snow-accumulation studies in Antarctica with ground-penetrating radar using 50, 100 and 800MHz antennafrequencies. Ann. Glaciol., 37, 194–198.

Watts, R.D. and A.W. England. 1976. Radio-echo sounding oftemperate glaciers: ice properties and sounder design criteria.J. Glaciol., 17(75), 39–48.

MS received 11 February 2005 and accepted in revised form 4 November 2005

Moore and Grinsted: Instruments and methods 163

VI

Is there evidence for sunspot forcing of climate at multi-year and

decadal periods?

John Moore,1 Aslak Grinsted,1,2 and Svetlana Jevrejeva3

Received 4 April 2006; revised 21 June 2006; accepted 25 July 2006; published 2 September 2006.

[1] It has been proposed that solar cycle irradiancevariations may affect the whole planet’s climate via thestratosphere, the Quasi-Biennial Oscillation (QBO) andArctic Oscillation (AO). We test this hypothesis byexamining causal links between time series of sunspotnumber and indices of QBO, AO and ENSO activity. Weuse various methods: wavelet coherence, average mutualinformation, and mean phase coherence to study the phasedynamics of weakly interacting oscillating systems. Allmethods clearly show a cause and effect link betweenSouthern Oscillation Index (SOI) and AO, but no linkbetween AO and QBO or solar cycle over all scales frombiannual to decadal. We conclude that the 11-year cyclesometimes seen in climate proxy records is unlikely to bedriven by solar forcing, and most likely reflects othernatural cycles of the climate system such as the 14-yearcycle, or a harmonic combination of multi-year cycles.Citation: Moore, J., A. Grinsted, and S. Jevrejeva (2006), Is

there evidence for sunspot forcing of climate at multi-year and

decadal periods?, Geophys. Res. Lett., 33, L17705, doi:10.1029/

2006GL026501.

1. Introduction

[2] There is considerable dispute as to the strength of thesun-climate link. Many proxy climatic time series exhibitsignificant power in the 11–14 year band, that severalauthors have been tempted to ascribe to solar sunspotcycles. However, detailed statistical analysis of many ofthese correlations shows them to be spurious or statisticallyinsignificant [Laut, 2003; Tsiropoula, 2003]. Decadal cyclesare fairly ubiquitous across the planet, and are thereforepersuasive of a global-scale climate mechanism [Jevrejevaet al., 2004]. Various mechanisms have been proposed toamplify the rather weak (0.1%) changes in solar energyoutput over an 11–12 year solar cycle to a level sufficientto produce changes in weather and climate. Several ofthese amplifying factors rely on the higher variability ofsolar energy at UV wavelengths to induce changesin stratospheric ozone and temperature, which can thenpropagate down to the troposphere [e.g., Baldwin andDunkerton, 2005; Labitzke, 2005].[3] The main features of the planet’s climate are the

ENSO and the polar annular modes. The strength of thepolar stratospheric vortex determines the index of annularmode, which are called the Arctic Oscillation, (AO) and the

Antarctic Annular Mode (AAM) [Thompson and Wallace,1998]. The QBO (quasi-biennial oscillation) is a quasi-periodic oscillation of the equatorial zonal wind betweeneasterlies and westerlies in the tropical stratosphere with amean period of 28 months. Almost all plausible sun-climatelinks rely on modification of the polar stratosphere, usuallywith some mediation role being played by the QBO.Labitzke [2005] summarize the possible influence of solarcycle on QBO. Kuroda and Shibata [2005] modeled theimpact of solar cycle on the AAM using a coupled chem-istry-climate model in two 21-year long model runs withconstantly repeating Sea Surface Temperature (SST). Theyfound that increased ultra-violet radiation led to a morepersistent signal from the AAM in the Antarctic stratospherethan during low UV model runs due to formation of anozone anomaly (amounting to 2–3%). Furthermore theyshow that it is UV rather than cosmic rays that produce thedifference in their model. Barnston and Livezey [1989], andlater Hameed and Lee [2005] showed that stratosphericperturbations are more likely to penetrate to the troposphereduring solar cycle maximum than minima, and that theeffect is also dependent on the direction of the zonal winddirection in the tropics. However these analyses rely only ondata available from 1948 and hence are not very statisticallysignificant. Kodera and Kuroda [2002] interpreted re-analyses data from 1979 to 1998 and proposed a mechanismfor the dynamical and radiative forcing of the stratosphereby the solar cycle, but there must be doubt to its statisticalrobustness as the data span less than two whole solar cycles.While it is clear that stratospheric anomalies can penetratedownwards to the troposphere, it is a rather atypicalphenomena [Baldwin and Dunkerton, 1999, 2001], and ingeneral the troposphere drives the stratosphere. Mayr et al.[2006] discuss a model simulation of the solar cycle and theQBO, and present evidence of a weak link, but theirmodeled solar cycle is fixed in period and amplitude.However, it is clear from both observational and modelingstudies that the stratosphere can provide an efficient and fasttransport mechanism for linking tropical and polar climate[Baldwin and Dunkerton, 2005; Jevrejeva et al., 2004].Thus the stratosphere provides a bridge between the annularmodes and ENSO phenomena, and so we may expect it beone factor that it is especially sensitive to the solar cycle.[4] In this paper we examine the plausibility of the

argument that solar cycles are significant factors in climateon multi-year and decadal timescales. Causality relation-ships are analyzed using wavelet coherence methods. Wave-let coherence is useful as relative phase relationshipsbetween two time series across a wide spectrum of temporalscales are produced. If the variable represented by one of thetime series is really the causal agent of the variability in thesecond time series, then a change in the first must always

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1Arctic Centre, University of Lapland, Rovaniemi, Finland.2Also at Department of Geophysics, University of Oulu, Oulu, Finland.3Proudman Oceanographic Laboratory, Liverpool, UK.

Copyright 2006 by the American Geophysical Union.0094-8276/06/2006GL026501$05.00

L17705 1 of 5

precede a reaction in the second. We firstly examine the linkbetween the QBO and the AO to verify if there is a coherentrelationship between these indices. Secondly we examine ifthere is a causal relationship between sunspot numbersand AO. Thirdly we test the mechanism that the solar cycledrives global climate via the polar regions by testingwhether the arctic leads tropical climate in the solar cycleband.

2. Methods and Data

[5] We use monthly time series of the AO [Thompsonand Wallace , 1998] and the QBO from http:/ /www.cdc.noaa.gov/Correlation/qbo.data. ENSO time seriescomes from monthly SOI [Ropelewski and Jones, 1987],and the monthly Nino 3.4 SST index [Smith and Reynolds,2004], defined as the monthly SST averaged over theeastern half of the tropical Pacific (5�S–5�N, 120�–170�W). We use the monthly International Sunspot numbersas the measure of the solar cycle (SC) available from http://sidc.oma.be/DATA/monthssn.dat, as accurate measurementsof total and surface solar irradiance variations have beenmade for only 2–3 decades. We removed the mean monthlyvalues (the annual cycle) from all series, and as the timeseries are all of different lengths, we restrict analyses to thecommon period of 1900–2000.[6] The methods we use in this paper rely on applying the

Continuous Wavelet Transform (CWT) to time series. Twouseful wavelets are the Morlet, defined as

y0 hð Þ ¼ p�1=4eiw0he�12h2 ; ð1Þ

and the Paul [Torrence and Compo, 1998]:

y0 hð Þ ¼ 2mim!ffiffiffiffiffiffiffiffiffiffiffiffiffiffip 2mð Þ!p 1� ihð Þ� mþ1ð Þ; ð2Þ

where w0 is dimensionless frequency and h is dimensionlesstime, and m is the order, taken as 4 here. The idea behind theCWT is to apply the wavelet as a band pass filter to the timeseries. The wavelet is stretched in time, t, by varying itsscale (s), so that h = s�t, and normalizing it to have unitenergy. The Morlet wavelet (with w0 = 6) provides a goodbalance between time and frequency localization and is agood choice for feature extraction so will be used in thewavelet coherence analysis. For broad band pass filteringapplications, we use the Paul as this is much less localizedin frequency space. The CWT of a time series X, {xn, n = 1,. . ., N} with uniform time steps dt, is defined as theconvolution of xn with the scaled and normalized wavelet.

WXn sð Þ ¼

ffiffiffiffidts

r XNn0¼1

xn0y0 n0 � nð Þ dts

� �: ð3Þ

The complex argument of WnX(s) can be interpreted as the

phases of X{f1 . . ., fn},[7] Following Grinsted et al. [2004] we define the

wavelet coherence of two time series X and Y{y1, . . . yn} as

R2n sð Þ ¼ S s�1WXY

n sð Þ� ��� ��2S s�1 WX

n sð Þ�� ��2� � S s�1 WY

n sð Þ�� ��2� ; ð4Þ

where S is a smoothing operator which is a Gaussian alongthe time axis and boxcar along the wavelet scale axis, anddesigned to fit the wavelet decorrelation length [Grinsted etal., 2004]. Notice that the definition of wavelet coherenceclosely resembles that of a traditional correlation coeffi-cient, and it is useful to think of it as a localized correlationcoefficient in time frequency space. We apply significancetesting using Monte Carlo methods using a red noise modelbased on the autocorrelation functions of the two time series[Grinsted et al., 2004].[8] The other methods we consider are non-linear inter-

actions between the two time series that may be chaotic.Both these methods rely on the phase expression of the timeseries derived from (3) with the Paul wavelet (2) of thedesired Fourier wavelength, l = 4ps/(2m + 1) [Torrence andCompo, 1998]. The broadband Paul wavelet allows signalsthat are relatively aperiodic to be included in the analysis,and it also makes the results we show very robust over alarge range of l. We utilize the average mutual information,I(X, Y ), between the two series, [e.g., Papoulis, 1984, chap.15]. In our case we are interested in causative relations, so itis appropriate to measure the I(X, Y ), between their respec-tive phases f, and q.

I X ; Yð Þ ¼ 1

log2 B

Xy2Y

Xx2X

p f; qð Þ log2p f; qð Þf fð Þg qð Þ ð5Þ

where p is the joint probability distribution function of Xand Y, and f and g are the marginal probability distributionfunctions of X and Y respectively, I is normalized by B, thenumber of histogram bins used to construct f and g.[9] Another measure of coherence between the two time

series is the angle strength of the phase angle differencebetween the series, also known as the mean phase coher-ence, r [Mokhov and Smirnov, 2006]:

r ¼ 1

N

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNt¼1

cos2 ft � qtð Þ þXNt¼1

sin2 ft � qtð Þvuut ð6Þ

We can search for the optimum relative phase delaybetween the two series by lagging one time series relativeto the other by a phase lag, D, in both (5) and (6).

3. Results

3.1. QBO-AO Relationship: Wavelet Coherence

[10] Figure 1 shows the wavelet coherence between thetwo time series. While both QBO and AO exhibit consid-erable power at biannual periods, there is no consistentphase relationship between the two series as clearly shownby the random orientations of the arrows in Figure 1. Thereis no evidence that supports the hypothesis that the strengthof any relationship varies with the solar cycle, as evenperiods of significant coherence at biannual periods sepa-rated by a decade do not show any consistent phaserelationship, e.g. the phase angle in 1960 was almost 180�different from that in 1970. There is also no evidence of anysignificant coherence in the solar cycle 11-year band.

3.2. Sunspot Numbers: Phase Coherence

[11] Having failed to establish any significant relation-ship between QBO and AO, we now assess if any other

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links between solar cycle and AO, which may not bedependent on the QBO, can be found. For this analysis weuse the I from (5) and r from (6). We firstly filter bothtime series with a Paul wavelet centred on 11 years. Wealso did the analysis with various other filters – the resultsare very insensitive and essentially the same as shown inFigure 2.[12] To illustrate the results expected from successful

application of the method we first examine the relationshipbetween SOI and Nino 3.4 (Figure 2a). It is clear thatthere is the expected obvious relationship, even when thedata are filtered with a 5 year centre frequency Paulwavelet. The peak in I and r is at 3 months with SOIleading Nino 3.4, as may be expected physically [Clarkeet al., 2000a, 2000b; Jevrejeva et al., 2004]. Figure 2bshows that there is no peak in I when D > 0 when sunspotnumber and AO are examined. The r has a pronouncedminimum around D = 0, where one may actually expectsignificant coherence if a simple physical relationshipexisted, but r does show a small peak at about D =7 years. However the low significance of the peak maybe judged by the much larger peak at D = �17 years,which is clearly unphysical as AO does not drive sunspotnumbers.[13] To show that the method is capable of detecting a

real relationship between two weakly interacting time series,we show the relationship between AO and Nino 3.4(Figure 2c), where there is clear relationship in the 14 yearband with a D = 3.5 years from tropics to Arctic. Jevrejevaet al. [2004] found a travel time of about 1 year for13.9 period year waves to travel from tropics to polarregions due to transmission by slow moving ocean wavesthat are detectable in global sea surface temperature field.The longer delay found here is likely a result of the method,which in the case of weakly coupled oscillators close tosynchrony, always displaces the peak to longer delays in lagspace [Cimponeriu et al., 2004].

[14] For completeness we also examined the sunspot -tropical climate relationship by examining its links with theSOI and Nino 3.4 indices (Figures 2d and 2e). Once againthere is no peak in I and low values of r for all D up to10 years.

4. Discussion

[15] It seems clear that there is no simple causativerelationship between sunspot numbers (and hence solarinsolation) and multi-year to decadal signals in the largecirculation systems that largely define the planet’s cli-mate. The non-linear analysis we have done also seems topreclude even quite indirect linkages between solar inso-lation and the SOI or AO, as we would still expect tosee evidence of sunspots cycles being ahead of AO evenif that delay was quite variable in absolute months oryears. More sophisticated analysis of phase interactionscould be done [e.g., Mokhov and Smirnov, 2006], how-ever such analyses are more suited to very low noiseoscillators, or to where such basic information as thesign of the phase is undetermined. If the solar cycle hasthe large impacts widely claimed then its influenceshould be readily seen in simpler analyses. In contrastthe 13.9 year cycle phasing is quite readily seen, despite

Figure 2. Relationships between time series that havebeen Paul wavelet filtered with centre frequency l,expressed as average mutual information, (I), (dottedcurves), and mean phase coherence, r, (solid line) as afunction of the phase lag (D) between the series for (a) SOIand Nino3, (b) AO and SC, (c) AO and Nino3, (d) SOI andSC and (e) Nino3 and SC.

Figure 1. Squared wavelet coherence between AO andQBO. The 5% significance level against red noise is shownas a thick contour. The relative phase relationship is shownas arrows (with in-phase pointing right, anti-phase pointingleft, and QBO leading AO by 90� pointing straight down).

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it contributing only about 5% of total variance to theSOI, Nino 3.4 and AO indices [Jevrejeva et al., 2004].Therefore we may confidently assert that the sunspotdriving of the AO is certainly less than 5% of totalvariance; any signal probably accounts for less variancethan the plausible limitations of noise in the long atmo-spheric circulation time series caused by varying spatialcoverage, or temperatures being used as proxies forpressure fields.[16] In time series spanning only 1 or 2 solar cycles it

is simply not possible to find significant phase relation-ships, Figure 1 also illustrates this important point - withonly 60 years of data there is only about 20 yearsof data in the decadal band which are not affected bythe data boundaries. While the details of the influenceof data boundaries are method-dependent, it is a com-mon feature of the short series regularly used in solar-climate analyses that deriving any statistical significance,especially in realistic red-noise backgrounds, is verychallenging.[17] There is direct evidence from both linear wavelet

coherence and from non-linear measures of interactionsbetween time series, that it is the tropics that lead polarclimate variability rather than vice-versa as has been pro-posed for a solar climate driver mechanism. Previousauthors have reported numerous examples of 11-yearcycles in many different proxy climate records, and oftenthey were assigned to the sunspot cycle. But in theanalyses that we have done we find no consistent phaserelationship between sunspot numbers and variables suchas sea ice extent or spring ice break-up in seas and ports,sea surface temperatures, sea level pressure, and variouslong meteorological records from cities in Europe. Many ofthe reports of 11-year cycles come from pure frequencydomain analyses such as Fourier transforms, and many ofthem have dubious levels of significance – either beingstatistically untested, or only tested against white-noiserather than an appropriate red noise background. In severalof these series there is some 11–14 year periodicitythat may actually be the 13.9 year signal discussed byJevrejeva et al. [2004] and which seems to be consistentlypresent from the tropics to the polar regions in sea surfacetemperatures [Jevrejeva et al., 2004], and in various proxyclimate series from ice cores [e.g., Fundel et al., 2006].However, there are other plausible mechanisms for an11-year cycle. A 5.2–5.7 year cycle is commonly ob-served in many climate records such as the AO [Jevrejevaand Moore, 2001], SSTs from several oceans [Unal andGhil, 1995; Moron et al., 1998], and central Englandtemperatures [Plaut et al., 1995]. A simple doubling ofthis period comes to 11 years, and such a doubling wouldbe frequently seen, especially in simple Fourier analysisof the data. Natural auto-correlated red-noise in the climatewould make this harmonic signal apparently more signifi-cant against white-noise background significance tests,thereby leading to claims for an 11 year signal rather thanthe 5.5 year signal.

5. Conclusion

[18] Numerous authors have considered the apparentlyself-evident hypothesis that since the sun is the fundamental

driving force for the earth’s climate, there should be clearlinks between the main climate patterns and the main indexof solar variability. However, rigorous testing of causativelinks between sunspots and climate indices finds no links ontime scales up to about 15 years. Solar driving of climatemust be present at timescales relevant to glacial-interglacialcycles and most-likely at shorter scales as well, but solarand climate proxies that meet length and resolution criterianecessary to prove the hypothesis are yet to be adequatelytested.

[19] Acknowledgment. Financial support came from the Thule Insti-tute and the Academy of Finland.

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Tsiropoula, G. (2003), Signatures of solar activity variability in meteorolo-gical parameters, J. Atmos. Sol. Terr. Phys., 65, 469–482.

Unal, Y. S., and M. Ghil (1995), Interannual an interdecadal oscillationpatterns in sea level, Clim. Dyn., 11, 255–278.

�����������������������A. Grinsted and J. Moore, Arctic Centre, University of Lapland, Box 122,

FIN-96101 Rovaniemi, Finland. (ag@glaciology.net; john.moore@ulapland.fi)S. Jevrejeva, Proudman Oceanographic Laboratory, Joseph Proudman

Building, 6 Brownlow Street, Liverpool L3 5DA, UK. (sveta@pol.ac.uk)

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VII

Oceanic and atmospheric transport of multiyear El Nino–Southern

Oscillation (ENSO) signatures to the polar regions

S. Jevrejeva,1 J. C. Moore,2 and A. Grinsted2,3

Received 29 June 2004; revised 15 October 2004; accepted 1 December 2004; published 24 December 2004.

[1] Using Monte-Carlo Singular Spectrum Analysis(MC- SSA) and Wavelet Transform (WT) we separatestatistically significant components from time series anddemonstrate significant co-variance and consistent phasedifferences between ice conditions and the Arctic Oscillationand Southern Oscillation indices (AO and SOI) at 2.2, 3.5,5.7 and 13.9 year periods. The 2.2, 3.5 and 5.7 year signalsdetected in the Arctic are generated about three monthsearlier in the tropical Pacific Ocean. In contrast, we show thatthe 13.9 year signal propagates eastward from the westernPacific as equatorial coupledwaves (ECW, 0.13–0.15ms�1),and then as fast boundary waves (1–3 ms�1) along thewestern margins of the Americas, with a phase difference ofabout 1.8–2.1 years by the time they reach the Arctic. Ourresults provide evidence of dynamical connections betweenhigh latitude surface conditions, tropical ocean sea surfacetemperatures mediated by tropical wave propagation, thewintertime polar vortex and the AO. INDEX TERMS: 1620

Global Change: Climate dynamics (3309); 3309 Meteorology and

Atmospheric Dynamics: Climatology (1620); 0312 Atmospheric

Composition and Structure: Air/sea constituent fluxes (3339, 4504);

4504 Oceanography: Physical: Air/sea interactions (0312); 4522

Oceanography: Physical: El Nino. Citation: Jevrejeva, S., J. C.

Moore, and A. Grinsted (2004), Oceanic and atmospheric transport

of multiyear El Nino–Southern Oscillation (ENSO) signatures to

the polar regions, Geophys. Res. Lett., 31, L24210, doi:10.1029/

2004GL020871.

1. Introduction

[2] Several studies indicate that the relationships betweenthe El Nino-Southern Oscillation (ENSO) and climateanomalies in the North Atlantic (NA) sector are both weakand ubiquitous, however most evidence of linkage comesfrom model simulations [e.g., Trenberth et al., 1998; Merkeland Latif, 2002] or from 50 years of reanalysis data [Riberaand Mann, 2002]. What is missing is clear observationalevidence, especially spanning long time periods. Impacts ofENSO are more plausibly seen in the NA sector duringwinter than summer, though the signal to noise ratiois rather low [Trenberth et al., 1998; Huang et al., 1998;Pozo-Vazquez et al., 2001]. These considerations motivatedus to use the NH time series of ice conditions and advancedstatistical methods to extract signals associated with ENSO.Our idea is that ENSO signals can be extracted from the iceconditions time series, since ice extent is an integrated

parameter of winter seasons and acts as a non- linear filterfor 2–13 year oscillations during the winter seasons[Jevrejeva and Moore, 2001]. ENSO quasi-biennial (QB)and quasi- quadrennial (QQ) signals were detected byMonte Carlo Singular Spectrum Analysis (MC-SSA) intime series of the Northern Hemisphere (NH) annular mode(NAM) and ice conditions in the Baltic Sea [Jevrejeva andMoore, 2001], which are consistent with results obtained forthe time series of sea ice cover from the Arctic and Antarctic[Gloersen, 1995; Venegas and Mysak, 2000]. This paperfocuses on the dynamic linkages, and putative mechanisms,especially for the low frequency component (13.9 yearsignal), between the ENSO and NAM circulation indicesand between the NH ice conditions and polar/tropical circu-lation patterns over the past 150 years using the MC-SSAand crosswavelet and wavelet coherence methods.

2. Data

[3] We used a 135-year time series of the monthly SOI[Ropelewski and Jones, 1987], as the atmospheric compo-nent of ENSO and the Nino3 SST index (1857–2001)[Kaplan et al., 1998], defined as the monthly SST averagedover the eastern half of the tropical Pacific (5�S–5�N, 90�–150�W) as the oceanic part. To ensure the robustness of theresults, we repeated the analyses for the monthly Nino1+2(0�–10�S, 90�–80�W) and monthly Nino3.4 (5�S–5�N,170�–120�W) time series [Kaplan et al., 1998]. NAMwas represented by the monthly AO index based on thepressure pattern (1899–2001) [Thompson and Wallace,1998] and the extended AO index (1857–1997) based onthe pattern of surface air temperature anomalies [Thompsonand Wallace, 1998].[4] Measures of winter season severity come from the

time series of maximum annual ice extent in the Baltic Sea(BMI), 1857–2000 [Seina and Palosuo, 1996], April iceextent in the Barents Sea (BE: Eastern part 10�–70�E; andBW: Western part 30�W–10�E), 1864–1998 [Venje, 2001]and the date of ice break-up at Riga since 1857 [Jevrejevaand Moore, 2001]. Global SST since 1854 were taken fromthe Extended Reconstructed Sea Surface Temperature(ERSST) [Smith and Reynolds, 2003] 2� � 2� datasetdown-sampled to yearly values.

3. Methods

[5] The analysis is performed using MC-SSA [Allen andSmith, 1996] and wavelet transform (WT) with a Morletwavelet [Foufoula-Georgiou and Kumar, 1995]. We appliedMC-SSA to calculate the contribution from non-linear long-term trends and quasi-regular oscillations to total variance,and analyse their development over time. We used the WTto determine both the dominant modes of variability and

GEOPHYSICAL RESEARCH LETTERS, VOL. 31, L24210, doi:10.1029/2004GL020871, 2004

1Proudman Oceanographic Laboratory, Birkenhead, UK.2Arctic Centre, University of Lapland, Rovaniemi, Finland.3Also at Department of Geophysics, University of Oulu, Oulu, Finland.

Copyright 2004 by the American Geophysical Union.0094-8276/04/2004GL020871$05.00

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how those modes vary in time. Statistical significance wasestimated against a red noise model. We calculated acoherence measure of the intensity of the covariance ofthe two series in time-frequency space [Torrence andWebster, 1999]. Phase difference between the componentsof the two time series was estimated as the circular mean ofthe phase for those regions with higher than 5% statisticalsignificance and which are outside the cone of influence[Jevrejeva et al., 2003].

4. Results

[6] Decomposition of time series of SOI, Nino3, AO andice conditions are presented in Table 1. Signals with 2.2,3.5, 5.7 and 13.9 year periodicities were detected in iceconditions time series and associated with similar signals inSOI/Nino3. Results are consistent with QB and QQ detectedby [Gloersen, 1995; Venegas and Mysak, 2000] in shortertime series of ice conditions in polar regions and withsignals in SST and sea level pressure (SLP) anomalies overthe Pacific Ocean [Huang et al., 1998; Torrence andWebster, 1999; White and Tourre, 2003; Ribera and Mann,2002]. As expected, the signals, detected in ice time series,are rather weak; with 1–6% contribution to the totalvariance, however, this amounts to 20–25% of all statisti-cally significant signals (Table 1).[7] To investigate the direct link between the localised

signals from atmospheric circulation index and ice conditionswe use crosswavelet and coherency methods. Results forthe Barents Sea ice conditions with AO/Nino3 (Figures 1a

and 1c) andBaltic SeawithAO/SOI (Figures 1b and 1d) showthat the phase lag where coherency >0.8 (corresponding tothe 95% significance level) is consistently about 3 monthsfor the 2.2–5.7 year signals. The same lag has also beenfound for the NA region [Pozo-Vazquez et al., 2001].[8] Wavelet coherence between SOI, Nino3, AO and

ice conditions is significant in the 12–20 year band. UsingMC-SSAwe isolated the 13.9 year oscillations in time series(Figure 2). The phase relationship between the oscillationswas then determined by cross spectral density estimates(relative to SOI). The 13.9 year SOI signal leads the AO by1.7 years and ice conditions by 1.8–2.1 years. We alsofound a 1.4 year lag between the 13.9 year signals from SOIand Nino3. The different phase relationship between signalswith periodicities of 2.2–5.7 and 13.9 years suggestsdifferent mechanisms of signal propagation from the equa-torial Pacific Ocean to the polar regions for the 13.9 yearsignal and for the shorter period ones.[9] The mechanism linking between the QB oscillations

(2.2–3.5 years) signals detected in ice conditions time seriesand in tropical forcing, has been described by Baldwin andDunkerton [2001], as extra-tropical wave propagation,

Table 1. Contribution From Significant Components Detected by

MC-SSA in Time Series of SOI, Nino3, AO, and Ice Conditions to

the Total Variancea

Contribution (%)

Period (years)

2.2 3.5 5.7 13.9

SOI 3 27 18 4Nino3 3 7 4*AO 9 3 6Barents sea 1 2 4Baltic sea 5 3 4 3Riga 4 4 5 5aRows in bold indicate EOFs at 95% in an AR(1) red-noise model,

* indicate EOFs at 90% level, others are significant at 95% level in whitenoise models.

Figure 1. The wavelet coherency and phase difference between AO/ice extent in the Barents Sea (a). Contours are waveletsquared coherencies. The vectors indicate the phase difference a) horizontal arrow pointing from left to right signifies in-phase and an arrow pointing vertically upward means the second series lags the first by 90 degrees (i.e., the phase angle is270�); b) the same for the AO/ice extent in the Baltic Sea; c) the same for the Nino3/ice extent in the Barents Sea; d) thesame for the SOI/ice extent in the Baltic Sea.

Figure 2. 13.9 year signals from time series of SOI, iceextent in the Baltic Sea, AO, ice extent in the Barents Sea,time series of date of ice break-up at Riga, Nino1+2, Nino3,Nino3.4.

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affecting breakdown of the wintertime stratospheric polarvortices. Since the AO may be interpreted as a physicalphenomenon associated with the structure of the polarvortex and related changes in the stratospheric and tropo-spheric pressure fields and the stratospheric polar vortexaffects surface weather patterns, the QBO has an effect onhigh latitude weather patterns [Baldwin and Dunkerton,2001; Castanheira and Graf, 2003]. The marked similarityin the phase lag of the 5.7 year signal to that of QB signal inSOI and ice conditions strongly suggests a similar mecha-nism of propagation.[10] We hypothesize that the lag between SOI and Nino3

signals can be explained by propagation of ECW on adecadal scale [White et al., 2003; Capotondi and Alexander,2001], from off-equatorial wind stress curl and triggeredECW by reflected coupled Rossby waves (CRW). Weconsider Rossby waves as a solution to the problem ofadjustment of the geophysical fluid to large-scale perturba-tion, based on the conservation of angular momentum.Wang and Weisberg [1994] found that ECW propagateeastward across the Pacific at speed of 0.28 ms�1. Whiteet al. [2003] showed the existence of coupled wavesoccurring in response to Rossby wave reflection on thewestern boundary that propagate eastward with phasespeeds of about 0.12 ms�1 on decadal period scales. Thesewaves propagate slowly eastward along the equator in thePacific Ocean in concert with global SST/SLP waves. Toinvestigate these decadal-scale waves, on the assumptionthat they are identifiable in the 13.9 year signals found inthe SOI and Nino3 series, we analysed the phase differencebetween the 13.9 years signals from SOI, Nino3, Nino1+2and Nino3.4 calculated by MC-SSA (Figure 2). The resultsprovide evidence of an increase of phase difference for13.9 year signals with eastward propagation. Using theselags gives an eastward propagation speed of 0.13–0.17 ms�1 for 13.9 year signals along the equator, whichis in excellent agreement with the speed of decadal- periodECW described by White et al. [2003].[11] We have projected the 13.9 year signal extracted

from SOI on the SST anomalies in the Pacific (1854–1997)(Figure 3). The strength of correlation and the phase ofcorrelation maps illustrate a meridional V-shape pattern,which is symmetric along the equator, propagating eastwardacross the ocean, with a speed of 0.15 ms�1 interrupted inthe central tropical Pacific Ocean by the development ofanomalous warm/cool tongues. When ECW reach themargins of North and South America they transform tobaroclinic Kelvin boundary waves (KBW), propagatingpolewards [Enfield and Allen, 1980; Meyers et al., 1998],of which the most energetic waves can reach as far north asAlaska [Meyers et al., 1998]. They are thought to start nearthe equator with Kelvin wave-like structure, and evolvetowards a continental shelf wave structure at higher latitudes[Suginohara, 1981]. This mechanism is well-established forequatorial Kelvin waves. For ECW, the excitation ofboundary Kelvin waves may be due to a combination ofthe incoming ocean signal and forcing by coupled windstress at the boundary (Clark [1992] suggested that windstress becomes relatively more important at longer periods).[12] Fast poleward propagation of KBW (a few months)

is confirmed by the phase angle map (Figure 3b), where wecan trace the propagation along the western margin of South

and North America and also along the Arctic continentalshelf.[13] A noticeable feature of Figure 3 is the strong

13.9 year signal around the Antarctic, a similar 13.9 yearsignal was reported as a long-standing feature in 2000 yearice core record from the Antarctic [Fischer et al., 2004],which they ascribed to a mode of variability in the AntarcticCircumpolar Wave (ACW). However, the ACW has alsobeen thought of as an ENSO teleconnnection, and Parket al. [2004] argue strongly that it is a geographically phase-locked standing wave train linked to tropical ENSOepisodes. This ENSO-modulated quasi-stationary variabilityis not zonally uniform, rather, the strongest ENSO impactis consistently concentrated in the Pacific sector of theSouthern Ocean, as we observe in Figure 3.

5. Conclusion

[14] We provide evidence of ENSO influence on thewinter climate variability in NH during the last 150 yearsvia signals in the 2.2, 3.5, 5.7 and 13.9 year bands. Thecontribution from the signals to the total variance is rela-tively weak, varies considerably with time, but is statisti-cally significant. Phase relationships for the differentfrequency signals suggest that there are different mecha-nisms for distribution of the 2.2–5.7 year and the 13.9 yearsignals. The 2.2–5.7 year signals are most likely transmittedvia the stratosphere, and the AO mediating propagation ofthe signals, through coupled stratospheric and troposphericcirculation variability that accounts for vertical planetarywave propagation.[15] The delay of about two years in the 13.9 year signals

detected in polar region can be explained by the transittime of the 13.9 year signal associated with ECW (0.13–0.17 ms�1) propagation in the Pacific ocean, KBW (1–3 ms�1) propagation along the western margins of theAmericas and by poleward-propagating of atmosphericangular momentum [Dickey et al., 2003]. This mechanismis supported by similar features in the Pacific sector of theAntarctic SST field.[16] Our results highlight the importance of tropical

variations for the Arctic and NA climate and probably atleast the Pacific sector of the Antarctic, suggesting a globalmode of interaction between atmosphere and ocean andconsistent with GCM experiments of a proposed ENSO-NA

Figure 3. Map of magnitude of correlation between SSTand the in-phase and quadrature 13.9 year SOI cycle, 95%significance is delineated by the black line (a). Map ofrelative phase angle (degrees) between the SST and in-phaseand quadrature SOI 13.9 year cycle (b).

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link [e.g., Trenberth et al., 1998; Dong et al., 2000; Merkeland Latif, 2002].

[17] Acknowledgments. Financial assistance was provided by NERCand the Thule Institute. Software is available at http://www.pol.ac.uk/home/research/waveletcoherence. We thank two anonymous referees.

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Baldwin, M. P., and T. J. Dunkerton (2001), Stratospheric harbingers ofanomalous weather regimes, Science, 294, 581–584.

Capotondi, A., and M. A. Alexander (2001), Rossby waves in the tropicalNorth Pacific and their role in decadal thermocline variability, J. Phys.Oceanogr., 31, 3496–3515.

Castanheira, J. M., and H.-F. Graf (2003), North Pacific–North Atlanticrelationships under stratospheric control?, J. Geophys. Res., 108(D1),4036, doi:10.1029/2002JD002754.

Clark, A. J. (1992), Low-frequency reflection from a nonmeriodional east-ern ocean boundary and the use of coastal sea level to monitor easternPacific equatorial Kelvin waves, J. Phys. Oceanogr., 22, 163–183.

Dickey, J. O., S. L. Marcus, and O. Viron (2003), Coherent interannual anddecadal variations in the atmosphere-ocean system, Geophys. Res. Lett.,30(11), 1573, doi:10.1029/2002GL016763.

Dong, B.-W., R. T. Sutton, S. P. Jewson, A. O’Neil, and J. M. Slingo(2000), Predictable winter climate in the North Atlantic sector duringthe 1997–1999 ENSO cycle, Geophys. Res. Lett., 27, 985–988.

Enfield, D. B., and J. S. Allen (1980), On the structure and dynamics ofmonthly mean sea level anomalies along the Pacific coast of North andSouth America, J. Phys. Oceanogr., 10, 557–578.

Fischer, H., F. Traufetter, H. Oerter, R. Weller, and H. Miller (2004),Prevalence of the Antarctic Circumpolar Wave over the last two millenniarecorded in Dronning Maud Land ice, Geophys. Res. Let., 31, L08202,doi:10.1029/2003GL019186.

Foufoula-Georgiou, E., and K. Kumar (1995), Wavelets in Geophysics,373 pp., Elsevier, New York.

Gloersen, R. (1995), Modulation of hemispheric sea-ice cover by ENSOevents, Nature, 373, 503–505.

Huang, J., K. Higuchi, and A. Shabbar (1998), The relationship between theNorth Atlantic Oscillation and the ENSO, Geophys. Res. Lett., 25, 2707–2710.

Jevrejeva, S., and J. C. Moore (2001), Singular spectrum analysis of BalticSea ice conditions and large-scale atmospheric patterns since 1708,Geophys. Res. Lett., 28, 4503–4507.

Jevrejeva, S., J. C. Moore, and A. Grinsted (2003), Influence of the ArcticOscillation and El Nino-Southern Oscillation (ENSO) on ice conditionsin the Baltic Sea: The wavelet approach, J. Geophys. Res., 108(D21),4677, doi:10.1029/2003JD003417.

Kaplan, A., M. A. Cane, Y. Kushnir, A. C. Clement, M. B. Blumenthal, andB. Rajagopalan (1998), Analyses of global sea surface temperature1856–1991, J. Geophys. Res., 103, 18,567–18,589.

Merkel, U., and M. Latif (2002), A high resolution AGCM study of theEl Nino impact on the North Atlantic/European sector, Geophys. Res.Lett., 29(9), 1291, doi:10.1029/2001GL013726.

Meyers, S. D., A. Melsom, G. T. Mitchum, and J. J. O’Brien (1998),Detection of the fast Kelvin waves teleconnection due to El Nino South-ern Oscillation, J. Geophys. Res., 103, 27,655–27,663.

Park, Y.-H., F. Roquet, and F. Vivier (2004), Quasi-stationary ENSO wavesignals versus the Antarctic Circumpolar Wave scenario, Geophys. Res.Lett., 31, L09315, doi:10.1029/2004GL019806.

Pozo-Vazquez, D., M. J. Esteban-Parra, F. S. Rodrigo, and Y. Castro-Diez(2001), The association between ENSO and winter atmospheric circula-tion and temperature in the North Atlantic region, J. Clim., 14, 3408–3420.

Ribera, P., and M. Mann (2002), Interannual variability in the NCEPreanalysis 1948–1999, Geophys. Res. Lett., 29(10), 1494, doi:10.1029/2001GL013905.

Ropelewski, C. F., and P. D. Jones (1987), An extension of the Tahiti-Darwin Southern Oscillation Index,Mon. Weather Rev., 115, 2161–2165.

Seina, A., and E. Palosuo (1996), The classification of the maximum annualextent of ice cover in the Baltic Sea 1720–1995, Rep. Ser. 27, pp. 79–91,Finn. Inst. Mar. Res., Helsinki.

Smith, T. M., and R. W. Reynolds (2003), Extended reconstruction ofglobal sea surface temperatures based on COADS data (1854–1997),J. Clim., 16, 1495–1510.

Suginohara, N. (1981), Propagation of coastal-trapped waves at low lati-tudes in a stratified ocean with continental shelf slope, J. Phys. Oceanogr.,11, 1113–1122.

Thompson, D. W. J., and J. M. Wallace (1998), The Arctic Oscillationsignature in the winter geopotential height and temperature fields, Geo-phys. Res. Lett., 25, 1297–1300.

Torrence, C., and P. Webster (1999), Interdecadal changes in the ENSO-monsoon system, J. Clim., 12, 2679–2690.

Trenberth, K. E., W. Branstator, D. Karoly, A. Kumar, N. Lau, andC. Ropelewski (1998), Progress during TOGA in understanding andmodeling global teleconnections associated with tropical sea surfacetemperatures, J. Geophys. Res., 103, 14,291–14,324.

Venegas, S. A., and L. A. Mysak (2000), Is there a dominant timescale ofnatural climate variability in the Arctic?, J. Clim., 13, 3412–3434.

Venje, T. (2001), Anomalies and trends of sea ice extent and atmosphericcirculation in the Nordic Seas during the period 1864–1998, J. Clim., 14,255–267.

Wang, C., and R. H. Weisberg (1994), On the ‘‘slow mode’’ mechanism inENSO- related coupled ocean-atmosphere models, J. Clim., 7, 1657–1667.

White, W. B., and Y. M. Tourre (2003), Global SST/SLP waves duringthe 20th century, Geophys. Res. Lett., 30(12), 1651, doi:10.1029/2003GL017055.

White, W. B., Y. M. Tourre, M. Barlow, and M. Dettinger (2003), A delayedaction oscillator shared by biennial, interannual, and decadal signalsin the Pacific Basin, J. Geophys. Res., 108(C3), 3070, doi:10.1029/2002JC001490.

�����������������������A. Grinsted and J. C. Moore, Arctic Centre, University of Lapland, FIN-

96101 Rovaniemi, Finland.S. Jevrejeva, Proudman Oceanographic Laboratory, Birkenhead CH43

7RA, UK. (sveta@pol.ac.uk)

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VIII

Nonlinear trends and multiyear cycles in sea level records

S. Jevrejeva,1 A. Grinsted,2,3 J. C. Moore,2 and S. Holgate1

Received 16 August 2005; revised 19 May 2006; accepted 12 June 2006; published 12 September 2006.

[1] We analyze the Permanent Service for Mean Sea Level (PSMSL) database of sealevel time series using a method based on Monte Carlo Singular Spectrum Analysis(MC-SSA). We remove 2–30 year quasi-periodic oscillations and determine the nonlinearlong-term trends for 12 large ocean regions. Our global sea level trend estimate of2.4 ± 1.0 mm/yr for the period from 1993 to 2000 is comparable with the 2.6 ± 0.7 mm/yrsea level rise calculated from TOPEX/Poseidon altimeter measurements. However, weshow that over the last 100 years the rate of 2.5 ± 1.0 mm/yr occurred between 1920 and1945, is likely to be as large as the 1990s, and resulted in a mean sea level rise of 48 mm.Weevaluate errors in sea level using two independent approaches, the robust bi-weightmean and variance, and a novel ‘‘virtual station’’ approach that utilizes geographic locationsof stations. Results suggest that a region cannot be adequately represented by a simple meancurve with standard error, assuming all stations are independent, as multiyear cycleswithin regions are very significant. Additionally, much of the between-region mismatcherrors are due to multiyear cycles in the global sea level that limit the ability of simple meansto capture sea level accurately. We demonstrate that variability in sea level records overperiods 2–30 years has increased during the past 50 years in most ocean basins.

Citation: Jevrejeva, S., A. Grinsted, J. C. Moore, and S. Holgate (2006), Nonlinear trends and multiyear cycles in sea level records,

J. Geophys. Res., 111, C09012, doi:10.1029/2005JC003229.

1. Introduction

[2] The rate of global mean sea level rise over the lastdecade (1993–2000) determined from TOPEX/Poseidonaltimeter measurements is 2.6 ± 0.7 mm/yr [White et al.,2005], which is significantly larger than estimates (1–2 mm/yr) of the 20th century linear trend for sea level rise[Church et al., 2001], and has reignited the discussion onwhether sea level rise is accelerating. Long term sea levelrates come from sea level gauge stations, that now numberin excess of 1000. However, most of these stations wereestablished in the 1950s, with only a handful extendingback 100 years or more. According to White et al. [2005],decadal trends in the global reconstruction of sea levelvary from 0 to 4 mm/yr during the period 1950–2000,with a maximum during the 1970s. However, Holgateand Woodworth [2004] argue that in 10-year trends since1950, the highest rates of sea level rise occurred in the1990s, with the lowest rates in the 1980s.[3] In this paper we address the development of the

regional and global nonlinear sea level trends over the fulllength of tide-gauge records, however as pre-20th centuryrecords are sparse and geographically very limited, we focuson the 20th century. We use several novel approaches incalculating mean sea level and errors for ocean regions andthen analyze the sea level using Monte-Carlo SingularSpectrum Analysis (MC-SSA) with confidence intervals

for nonlinear trends [Allen and Smith, 1996; Moore et al.,2005]. This approach provides information on changes incontribution from 12 large ocean regions to global sea levelrise over time. In addition, we use wavelet and waveletcoherence methods [Torrence and Compo, 1998; Grinsted etal., 2004] to investigate the role of signals with periods of 2–13.9 years in the variability of sea level records, their links tothe large scale atmospheric circulation patterns and howthese relationships have changed over the past 100 years.[4] In previously published results [see, e.g., Church et al.,

2001; Nerem and Mitchum, 2002], linear trends were calcu-lated by ordinary least squares methods, which can producemisleading results if the residuals do not have a white noisespectrum [Kuo et al., 1990]. Additionally, most tide gaugerecords demonstrate distinctly nonuniform trends, whichmakes the linear trends sensitive to the arbitrarily chosenstart and end dates. Here we apply MC-SSA to extractnonlinear long-term trends from time series of mean sealevel. We filter the low frequency variability by removing thestatistically significant 2–13.9 year oscillations associatedwith large scale atmospheric circulation [Unal and Ghil,1995; Jevrejeva et al., 2005]; along with multidecadaloscillations (14–30 year periods). We then estimate thenonlinear long-term trend, providing the statistical signifi-cance of the trend, and its confidence intervals.

2. Data and Methodology

2.1. Tide Gauge Stations

[5] We aimed to use all relative sea level (RSL) monthlymean time series in the Permanent Service for Mean SeaLevel (PSMSL) database [Woodworth and Player, 2003].However, data from Japan were excluded from the analysis

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due to uncertainty in earthquake-related land movement ofbench marks and tide gauge stations. Detailed descriptionsof the RSL time series are available from www.pol.ac.uk/psmsl. No inverted barometer correction was applied. RSLdata sets were corrected for local datum changes and glacialisostatic adjustment (GIA) of the solid Earth [Peltier, 2001].In the GIA data set there were missing rates for severalstations particularly in Arctic Siberia, and much of Aus-tralia. We addressed these deficiencies in GIA by assigninga station with no GIA correction the same GIA as thenearest station within 100 km of it. If no GIA correctedstation was within 100 km, we used an interpolation schemein assigning GIA rates from stations up to 1000 km away.Stations further than 1000 km from sites with assigned GIAwere removed from the analyses.[6] There is no common reference level for the tide gauge

records and this provides a problem when stacking recordsthat do not cover the same time periods. One way overcomethis problem is to calculate the rate of change in sea level foreach station and stack the rates [Barnett, 1984]. However,many stations have historically only been measured forsome months of the year and an annual cycle in sea levelcould therefore lead to severe bias. To maximize data usage,we calculate the mean annual rate for a given month over awhole year (e.g., the rate in January is calculated as the Julyto July difference). Using this method removes all system-atic sub-annual signals from the data. Data gaps shorter thanone year in the final rate series are filled by interpolation.This procedure results in data from 1023 stations containing385324 individual monthly records, of which 22130 monthswere filled by interpolation. The maximum number ofstations in any year is 585, with only 70 stations in 1900,and 5 in 1850. Due to the time lag between data collectionand supply to the PSMSL recent decades have also seenreductions in station numbers with only 390 stations in2000.

2.2. Regional Trends

[7] The geographical distribution of tide gauges providesa poor sampling of the ocean basins, and most of tidegauges are situated in the Northern Hemisphere (Europe,Japan, and the United States). Additionally tide gaugerecords do not cover the same time period. It is useful todefine 12 sea level regions (Figure 1) and calculate themean sea level time series for the each region. The degree towhich a station represents fluctuations in sea level across aregion, and errors in the sea level measurements of stationsare key issues in estimating region sea level trends. We usetwo independent methods of estimating these. The first isthe standard method of the statistical bi-weight robust meanand variance [Mosteller and Tukey, 1977], commonly usedin dendrochronology to produce mean indices that arerobust in excluding time series that are far from the medianseries. This procedure leads to a mean and a standard errorestimate directly from all the stations in a region. Thesecond method we employ seeks to resolve the issue ofspatial inhomogeneity. The stations are not evenly distrib-uted spatially, and therefore we have developed a novel wayof stacking the stations. Dealing separately with each regionof the world ocean (Figure 1), we recursively find the twoclosest stations (x and y) and merge them creating a newvirtual station located halfway between the original stations.

The error for the virtual station for the ith year can be writtenas

ei ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

i þ r2i

q; ð1Þ

where m is the measurement error associated with eachstation measurement and r is the misfit error, that is thedifference in sea level trend recorded at the two stations inany particular year. The error e is treated as a measurementerror when a virtual station is merged with another station inthe next step in the recursion. Generally time series from xand y do not span exactly the same period. The meanmeasurement error (m) is then calculated as

mi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

x;i þ m2y;i

qni

; ð2Þ

where mx and my are the measurement errors of station x andy respectively. In years where only one station has data ni = 1and the other is simply left out of the equation, otherwiseni = 2. The misfit error tells how well the virtual stationrepresents stations x and y and is written

ri ¼ �xy

2ffiffiffiffini

p ; ð3Þ

where sxy is calculated as the standard deviation of the ratesat x minus those at y over the interval of common overlap.When the recursive procedure has been carried out we areleft with the sea level rate of the entire region and itsassociated error.[8] Figure 2 shows the yearly sea level rate of change in

the eastern North Atlantic region with the bi-weight meanand the virtual station method. The general features agree

Figure 1. Location of the tide gauges included in this study(12 regions: cpacific, Central Pacific; neatlantic, northeastAtlantic; nepacific, northeast Pacific; nwatlantic, north-west Atlantic; sepacifc, southeast Pacific; swatlantic,southwest Atlantic; wpacific, western Pacific; antarctic,Antarctica; arctic, Arctic; baltic, Baltic; indian, Indian;mediterr, Mediterranean).

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well over most of the full length of the record, though it canbe see that the bi-weight mean estimate tends to havesmaller amplitude in some periods where statistical outliersare removed from the robust mean. Note that there is verylittle overall trend apparent in the rate curve. The variationin the standard errors shows more interesting differencesbetween the methods. In the early part of the record, whenthere are few stations, the two methods give quite similarestimates of the errors. However, when the number ofstations becomes much larger in the middle of the 20thcentury, the bi-weight standard error becomes muchsmaller than the virtual station estimate. This is becausethe simple assumption of statistical independence inherentin the bi-weight method, reduces the standard error of themean by the square root of the number of stations.However, the virtual station error remains large as it isdominated by the station mismatch error in the final stepof the recursion process, when two far-apart stations thatgenerally have large mismatch, are meaned. This reflectsthe true limit of how representative a single curve is ofregion-wide sea level. Therefore the virtual station error isa much more reliable estimate of uncertainty than a simplerobust standard error.

2.3. Global Sea Level

[9] Previous authors have used estimates of the lineartrends for oceanic basins (found by averaging the basinrates) to compute a simple global trend by taking the meanand the error in the mean between basins [Douglas, 1997].Simple averaging of these errors would lead to incorrect

uncertainties in sea level rise. Despite the relatively largevirtual station error estimates, we find that integrating thetrends results in positive rates in all regions over the last150 years. This strongly suggests that errors in rates are notrandom, but caused by systematic oscillations in sea levelwithin the regions which will be manifested as autocorre-lation within the errors. We define a region mismatch error(bi) as the difference in sea level rates between that in theparticular region and the mean over all the regions in the ith

year. We can correct for the autocorrelated fraction of the bierror by considering its power spectrum (Figure 3). Thepower spectrum shows a marked decrease with frequency,due to the importance of multiyear oscillations in sea level.Since the low frequencies dominate the total mismatcherror, we should reduce the mismatch error estimate onthe yearly rates by the square root ratio of the highfrequency power to the low frequency power, (P). FromFigure 3 it can be seen that this ratio is typically about 2–3and so the between-region sea level yearly rate errorestimate should be only about 1/2 of the standard errorfound simply by considering the rates between regions. Tocalculate the total error in the global trend (gi) we need toadd (as a vector) this between-region error to the stationrepresentativity error (again correcting for the power spec-trum of the error).

gi ¼ P

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2i þ

Pj

e2ij

N2i

vuut ð4Þ

Figure 2. Comparison of the northeastern Atlantic sea level rates from monthly data (top and zoomedinset showing generally close agreement in the two estimates) and associated standard error (bottom)estimated using bi-weight robust means (black) and the virtual station method (grey) from equation (1).

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where Ni is the number of regions with stations operating inthe ith year, and eij is the ei from equation (1) in jth region.This is the error that we shall report in global rate estimates.

2.4. Sea Level Analysis

[10] In this study the sea level time series were consideredas a combination of the long-term trends, oscillations andnoise. For trend analysis we use a novel approach [Moore etal., 2005] that has significant advantages over traditionallow-pass filtering in that we can estimate the significance ofthe trend, its confidence intervals and its contribution to thetotal series variance. We decompose the time series usingthe MC-SSA [Allen and Smith, 1996] with data-adaptiveorthogonal filters, separating the original time series into thenonlinear low frequency trend, quasiperiodic signals andnoise, distributed among the filters. We use an embeddingdimension of equivalent to 30 years, which means lowfrequency oscillations will appear as trends in a similar wayto filtering with a Gaussian of 60 year cutoff period [Mooreet al., 2005]. We calculate the statistical significance ofcomponents reconstructed by convoluting the original serieswith each of the filters and comparing against red noisemodels (Monte Carlo method, with 1000 surrogate serieswith the same length, observed variance and lag-one auto-correlations as the original time series). All results presentedin this paper are statistically significant at the 95% levelagainst white noise models, and many are also significantagainst red noise models. The contribution from the trendsand quasi-regular oscillations to the total variance, arecalculated as a projected component relative to the originaltime series. Finally, we remove the effects of the shorttimescale variability (2–30 years) from the time series, thenextract the long-term trends and analyze their developmentover time. Mann [2004] discusses methods of calculating

trends close to the observational boundaries, and we use avariation on the minimum roughness criterion here. Weimplement the minimum roughness by extrapolating thetime series 30 years forward and backward in time using aleast squares linear fit onto the 30 years of data nearest to aboundary before applying the SSA nonlinear trend filter.The nonlinear trend at the data boundaries may therefore besaid to be composed of 30 data points and 29 extrapolatedpoints from the linear trend.[11] In comparison with the least squares linear fits, often

used to compute confidence intervals of means, we have theadvantage of requiring errors to conform only to the optimaldata adaptive filter specifying the trend, and thus find thatconfidence intervals in nonlinear time series are perhaps 3–4 times smaller than found by linear fitting [Moore et al.,2005].[12] Nonlinear trends for regional sea level yearly rate are

calculated from the integrated mean regional monthly ratesto give a sea level curve, then performing MC-SSA toextract the nonlinear trend and finally differentiating to givethe nonlinear trend in regional sea level rate. This methodavoids uncertainties in decomposing the monthly rate wherethe trend accounts for a small fraction of the variance in thedata, whereas using the integrated data the trend oftenaccounts for 50% of the variance.[13] Global sea level rate is calculated by simply averag-

ing the regional rates (the Baltic being excluded, as dis-cussed later). Global sea level is found from the mean of theregional sea level curves. The error in global sea level isestimated from the standard deviation of the regional sealevel curves for each year, however, the errors produced arenot statistically independent for similar reasons as thosediscussed earlier in relation to equation (4), and so weestimate the confidence interval using a ‘‘jack knife’’method [Press et al., 1992] which can compensate for thiseffect.[14] To investigate the role of 2–30 year oscillations we

use MC-SSA and wavelet methods. We apply the waveletmethod [Torrence and Compo, 1998] and decompose thetime series into time –frequency space, in order to deter-mine both the dominant modes of variability and how thosemodes vary in time. We compare results with the oscilla-tions detected by MC-SSA, and the consistency of theresults obtained by these independent methods strengthenstheir credibility.[15] To analyze the direct links between the statistically

significant components in regional sea level time series andtime series representing the large scale atmospheric circu-lation we use the wavelet coherence method [Grinsted etal., 2004; Jevrejeva et al., 2005]. We give descriptions ofthe changes in covariance between time series and showthat there is considerable linkage between the statisticallysignificant components in time series of sea level andatmospheric circulation.

3. Results

3.1. Nonlinear Trends in Sea Level

[16] Figure 4 shows the rates of the nonlinear trends in 10of the 12 regions (Antarctic and southeastern Pacific, timeseries are too short to perform trend analysis). Figure 5shows the mean global sea level and its nonlinear trend with

Figure 3. Normalized square rooted power spectrum ofthe mismatch error between each of the regions (Figure 1),except the Baltic, and the mean global sea level rate,calculated using an order 6 maximum entropy method. Thethick black curve is the region mean spectrum. The powerspectrum peaks at low frequencies due to long periodoscillations, while the high frequency tail represents the‘‘correct’’ mismatch error.

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95% confidence interval, and also the nonlinear global sealevel rate. Only 5 stations were active during 1850 and all ofthem were in the North Atlantic or the Baltic (Figure 4), soestimates of global trends are rather uncertain before 1900,as is reflected in the large errors in Figure 5. The largeacceleration in sea level 1850–1870 coincides with a fewnew stations becoming active, and is probably not statisti-cally significant.[17] In contrast with linear trends, where the rate of mean

sea level rise is constant, our results (Figure 4) reveal theevolution of sea level rise during the 20th century and showthat the highest regional rates of up to 3–5 mm/yr occurredbetween 1920–1950 (with some regional variations, e.g.,the later maximum for the southwestern Atlantic andwestern Pacific). The major contributions to the global sealevel rise during 1920–1940 are from the northwesternAtlantic (4.2 ± 1.0 mm/yr), Indian (3.5 ± 1.0 mm/yr), andMediterranean (3.1 ± 1.0 mm/yr) regions. Even smoothedby the 30 year SSA window, the trends from the differentocean regions show slightly dissimilar patterns and stilldemonstrate some cyclic variability. This cyclicity is asso-ciated with longer term oceanic variations [e.g., Ghil, 2001],changes in thermal expansion and water mass adding to theocean, which may provide nonuniform regional sea levelrise, and melting of continental ice leads to the significantgeographic variations in the sea level change due to bothgravitational and loading effects [Mitrovica et al., 2001].[18] A notable feature of Figure 4 is that for the last

20 years the regional trends are more coherent in the sensethat the trends are closely grouped, and the rates for thesouthern and northern hemisphere become similar. The trendfor the Baltic Sea is strongly influenced by limited waterexchange with larger regions and fresh water flux variations[Meier and Kauker, 2003] which leads to its rate curve beingquite dissimilar from the other regions, and in our study wehave excluded the Baltic Sea time series from the calculationof the global sea level trend. Arctic sea level trend has beenaccelerating for the last 30 years, which can be explained by

the warming of the Arctic coupled with increased influx ofwarmer Atlantic waters to the arctic region, the increase offresh water inflow from Siberian rivers [Dickson et al., 2002;Wu et al., 2005], and changes in the Arctic Ocean circulation[Proshutinsky et al., 2001].[19] Applying a least squares linear regression analysis to

the global sea level in Figure 5 gives a trend of 1.8 mm/yrbetween 1900 and 2000, as found in earlier studies [Church

Figure 4. Rates of nonlinear sea level trends in the regions defined in Figure 1, found using an SSAembedding dimension equivalent to 30 years. Errors are not shown in the plot, but using the exampleof Figure 2 for northeastern Atlantic, the maximum error at the start of the time series would be about150P/

p(12 � 30 � 2) = 2–3 mm yr�1 and the minimum error about 1 mm yr�1; other regions will have

similar errors. Note that extrapolation at data boundaries means that curves are increasingly moreuncertain within 30 years of the start and end of the records.

Figure 5. (top) Nonlinear trend (thick black line) in globalsea level using an SSA embedding dimension equivalent to30 years. Paralleling the trend is the 95% confidenceinterval in the trend found by considering the mismatchbetween the regional sea level curves; thin curve is theyearly global sea level. (bottom) Rate of the global sea leveltrend, and its standard error from equation (4).

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et al., 2001]. However, our results show that global sea levelrise is irregular and varies greatly over time, it is apparentthat rates in the 1920–1945 period are likely to be as largeas today’s. Nevertheless, considerable uncertainties remain.[20] The nonlinear trend for the 1990s show about the

same sea level rise (2.4 ± 1.0 mm/yr) as TOPEX/Poseidon,which demonstrates the ability of the nonlinear trend toresolve (with certain errors) the temporal evolution of thespatial sea level field. The largest contributions to the globalsea level trend during the 1990s are from northwesternAtlantic, northeastern Atlantic and Arctic regions (Figure 4),which is in good agreement with results from Levitus et al.[2005] showing the warming of the Atlantic regions provid-ing the largest contribution to the global ocean heat contentduring 1955–2003.[21] Sea level is an integrated indicator of climate vari-

ability, which reflects changes in the dynamic and thermo-dynamic in atmosphere, ocean and cryosphere. Both landsurface [Jones and Moberg, 2003] and sea surface temper-ature data [Kaplan et al., 1998; Rayner et al., 2003] are ingood agreement with the gsl curve in Figure 5, in particular,the increases from 1910 to 1940, and from 1980 until thepresent, as well as a stagnation between 1940 and 1975.One possible explanation for the sea level accelerationduring 1920–1945 was the low level of volcanic activity.Recently published model results [Church et al., 2005]suggest that multiyear to decadal sea level variability issignificantly affected by volcanic activity. The climatesystem responds to volcanic forcing on a range of time-scales. The short period (generally lasting 1–3 yrs) responseis mainly due to stratospheric aerosols from explosivevolcanic eruptions scattering incoming solar short waveradiation, which leads to cooling of the global averagetemperature [Sato et al., 1993; Robock and Liu, 1994;Robock, 2000] and a reduction in rainfall [Robock andLiu, 1994; Gillett et al., 2004; Lambert et al., 2004]. Asimple comparison between the rate of sea level rise and

stratospheric aerosol optical thickness, the principal param-eter affecting stratospheric aerosol climate forcing [Sato etal., 1993], is shown in Figure 6 and suggests that rapid sealevel rise prior to 1980 occurred during periods of lowvolcanic activity. While it is tempting to perform regressionmodeling of sea level on the trends in optical depth and e.g.,global temperature, it is problematic as the short termvolcanic impact on global temperatures means that thevariables are highly co-dependent. Applying regression onMC-SSA trend data (effectively moving averaged with a60 year window) reduces the available degrees of freedomin 150 years of data to the point where fits are statisticallyinsignificant. However, our hypothesis is supported by theresults from the SI200 version of Goddard Institute forSpace Studies (GISS) simulations, where since 1950, min-ima in the ocean heat content occurred due to the largevolcanic eruptions of Agung, El Chichon and Pinatubo[Hansen et al., 2002]. The observed ocean heat content[Levitus et al., 2005] provides evidence of cooling afterall three eruptions (Agung, El Chichon, and Pinatubo).Bengtsson et al. [1999] also demonstrate that the Pinatuboeruption had a recognizable effect on climate, with directforcing limited in duration to a few years and a long-termeffect, lasting 7 years, due to cooling of the oceans.

3.2. Role of Oscillations in the 2–30 Year Band

[22] Using MC-SSA, we extract statistically significantcomponents from time series of regional sea level. Signalsin the 3.5–13.9 year band contribute from 5 to 20% ofvariability in time series. There is also a substantial contri-bution from signals with 13–30 year periodicity. In ourstudy signals with 13–30 year periodicity from regionaltime series are statistically significant against a white noisemodel. However, for individual stations these signals arestatistically significant against red noise [Unal and Ghil,1995]. Our results are in good agreement with oscillationsdetected at 3.5, 5.2–5.7, 7–8.5 and 10–13.9 years forindividual sea level station time series by Unal and Ghil

Figure 6. Rate of the nonlinear global sea level trend (thick black line, mm/yr) and stratospheric aerosoloptical depth (thin line) [Sato et al., 1993].

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[1995]. It is notable that 3.5–13.9 year oscillations demon-strate an increase in amplitude since the 1940s in severalregions: northeastern Atlantic, northwestern Atlantic, andeastern Pacific, presented in Figure 7.[23] To isolate the different timescales of variability, and

analyze the changes of variance in the time series of sealevel we examine their behavior in time-frequency spaceusing the Morlet wavelet. Figure 8 shows the wavelet powerspectrum of 4 regional sea level time series, displayed as afunction of cycle period and time. The left axis is theFourier period; the bottom axis shows the time in years.The strong nonstationary behavior of the spectra is clearlyseen. Most of the time series show an increase of power inthe wavelet power spectrum at 2–30 year periods since1940s.[24] Oscillations in the 2.2–7.8 year band in individual

sea level records have been associated with large scaleatmospheric circulation signals [Unal and Ghil, 1995;Jevrejeva et al., 2005], which generate a sea level responsethrough a number of processes: the direct influence ofchanged atmospheric pressure, changes in wind stresses,and changes in atmosphere-ocean fluxes as well as stormsurges. Previously published results have revealed theinfluence of large scale atmospheric circulation on sea levelvariability for various ocean basins. According to theHughes et al. [2003] Antarctic sea level is highly coherentwith the Antarctic Annular Mode (AAO). Substantial con-tributions from changes in large scale atmospheric circula-tion patterns to the sea level variability in Arctic were foundby Proshutinsky et al. [2001]. The role of the SOI index invariability in the Pacific sea level is discussed by Church etal. [2004].[25] To identify the frequency bands within which time

series of sea level and the large scale atmospheric circula-tion are co-varying, we use the wavelet coherency method[Grinsted et al., 2004; Jevrejeva et al., 2005]. Figure 9ademonstrates the influence of the Arctic Oscillation (AO)[Thompson and Wallace, 1998] on the variability of theNorth Atlantic associated with 2.2–13.9 year signals after1940, which is in excellent agreement with results for theindividual records reported by Jevrejeva et al. [2005].Influence of the North Atlantic Oscillation (NAO) [Jones

et al., 1997] in the Mediterranean region, shown by Tsimplisand Josey [2001], is associated with 2.2–7.8 year signals(Figure 9b). There is also evidence of a shift in the period ofmaximum coherence from the 2.2–3.5 to the 3.5–7.8 yearband for the northeastern Atlantic after 1950 and noticeablechanges in the 2.2–5.7 year band for the Mediterraneanregion.[26] Indian and Pacific Oceans sea level variability is

mainly associated with Southern Oscillation Index (SOI)[Ropelewski and Jones, 1987] signals at 2.2, 3.5, 5.7 yearperiods. Figures 9c and 9d demonstrate that the relationshipis not stationary and the influence of SOI has generallyincreased over the last 60 years over a broadening spectrumof periods. Our results (Figure 9d) show increasing influ-ence from 13.9 year signal, previously found in Sea SurfaceTemeprature (SST) in the Pacific Ocean and extending tothe polar regions [Jevrejeva et al., 2004]. We also use thePacific Decadal Oscillation (PDO) index [Zhang et al.,1997], derived as the leading mode of monthly SSTanomalies in the North Pacific Ocean, as a representationof the Pacific climate variability. The signature of the 2.2–30 year signals, associated with PDO, is clearly seen in theeastern Pacific sea level variability (not shown here). ThePDO’s influence since 1940 shows an increase in the lowfrequency power in the 13–30 year band, somewhat similarto seen for the SOI in Figure 9d, and large changes in the3.5–7.8 year band. Our results are consistent with aprevious study by Church et al. [2004] where SOI wasfound to be strongly anticorrelated (�0.78) with the firstEmpirical Orthogonal Function (EOF) of sea level in thePacific. Furthermore, increased variance is sea level isassociated with a tendency for ENSO events to be morefrequent, persistent and intense in the equatorial Pacific inthe last two decades.

4. Discussion

[27] Our global sea level trend estimate of 2.4 ± 1.0 mm/yrfor the period from 1993 to 2000matches the 2.6 ± 0.7 mm/yrsea level rise found from TOPEX/Poseidon altimeter data.However, Figure 5 suggests that during 20th century asimilar sea level rise was observed between 1920 and 1945.

Figure 7. (a) Standardized sum of the statistically significant (at the 95% confidence level against rednoise model) SSA components for the 3.5- and 2.4-year oscillations in time series of mean sea level innortheastern Atlantic region; (b) for 5.2-year oscillations in northeastern Pacific. Both showing increasedamplitude since 1950.

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The maximum rate of global mean sea level rise was 2.5 ±1.0 mm/yr during the period 1920–45, with cumulativerise of 48 mm in 25 years.[28] These results differ slightly from previously pub-

lished conclusions [e.g., Cazenave and Nerem, 2002]concerning historical rates of mean sea level rise duringrecent decades. Most probably the difference in the resultscan be explained by the use of different data (number ofstations, location of the stations), GIA corrections, methodsand errors for trend calculation. Church et al. [2004]pointed out that with decadal variability in the computedglobal mean sea level, it is not possible to detect asignificant increase in the rate of sea level rise over theperiod 1950–2000. However, results from White et al.[2005] suggest that decadal reconstructed sea level trendshave varied from 0 to 4 mm/yr since 1950. The results ofHolgate and Woodworth [2004] for the second half of the20th century reveal that the highest rates in decadal sea leveltrends were in the 1990s, but the lowest rises were in the1980s, demonstrating a significant variation in rates of sealevel rise.[29] Furthermore, observed ocean heat content, the major

contributor to sea level rise, has significant decadal vari-ability [Levitus et al., 2005]. According to results from

simulations using the GISS global climate model thesedecadal heat content changes do not appear to be causedby the climate forcing, but are more likely to be dynamicalfeatures [Hansen et al., 2002]. Similar results are found withthe National Centre for Atmospheric Research (NCAR)Parallel Climate Model [Barnett et al., 2001]. Multidecadalvariability in sea level records most probably links to thevariability in the heat transported by the thermohalinecirculation, driven by the temperature and salinity differ-ences [Deser and Blackmon, 1993; Rajagopalan et al.,1998; Rodwell et al., 1999]. Such changes in the thermalstructure of the ocean can result in steric (density) sea levelchanges [Knutti and Stocker, 2000].[30] Our results show that there is an increase in 2–30 year

variability is sea level over the recent decades for severalregions. We speculate that one of the possible sourcesfor increase in 2–13.9 year variability could be the largescale atmospheric circulation represented by the SOI,AO, NAO and PDO.[31] Figure 4 shows some evidence that the nonlinear

trends for the different regions are in closer agreementaround 1980 than previously, though it must be noted thaterrors in the trends are relatively large. AdditionallyFigure 8 shows that 2–30 year variability is increasing.

Figure 8. Wavelet power spectrum (Morlet) of mean monthly mean sea level for the (a) northeasternAtlantic; (b) northeastern Pacific; (c) Mediterranean; (d) Indian. Contours are in variance units. In allpanels the black thick line is the 5% significance level using the red noise model; solid line indicates thecone of influence. The color bar represents normalized variances. All of the time series show an increaseof power in the wavelet power spectrum at 2–30 year periods since 1940s as can be seen by the spread ofthe red and yellow shading to the upper rights of the panels.

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One possible explanation is a warmer ocean makes the largescales more coherent, as ocean temperature and heat contentincrease in all regions of the world. At the same time, the2–13.9 year variability in sea level records in some regions,for example, North Atlantic, Mediterranean Sea, easternPacific, and Indian Oceans associated with atmosphericforcing, which has increased over the last 50 years [Wakelinet al., 2003; Jevrejeva et al., 2005], shows the relationshipis changing with time. This phenomenon may be expectedon theoretical grounds [e.g., Tsonis, 2004]. It is likely as awarmer ocean leads to more large scale teleconnections,while also tending to introduce more variability at anyparticular place [Tsonis, 2004]. This can be seen inFigure 9d where long period signals increase in strengthand generally the Pacific sea level at all frequencies is seen tobe more coherent with SOI. Increases in high frequencycoherency in the Pacific (Figure 9d) are also suggestive ofthe increasing frequency of El-Nino events [Church et al.,2004]. Similar results are seen in many other records ofthe climate system, for example global temperature records[Jones and Moberg, 2003] and 500 mbar pressure height

fields [Tsonis, 2004]. The manifestation of the climaticmechanisms responsible for the increased long-range tele-connections is presently little understood, but must berelated to the low-order dynamics of the global system,for example the increasing rate of El-Nino events, andreduction in La Nina in a warming world discussed byTsonis et al. [2005].[32] Our study shows the importance of estimates of

uncertainties and hence some of the limitation of tidegauges measurements, previously pointed out by Churchet al. [2004] and Cazenave and Nerem [2004]. In ourmethod the estimates explicitly account for the spatialredistribution of sea level, and record the temporal variabil-ity better than estimations from individual stations. Theerrors in the estimates are inevitably quite large, due to thepoor special distribution of tide gauges, as they are locatedonly on the continental margins and ocean islands. Thealternative method of producing a mean sea level curve bychoosing tide gauges from suitable locations by selectivecriteria, as used by Douglas [1997], reduces the number ofuseful tide gauges to very few (25). Thus a producing

Figure 9. (a) Wavelet coherency between AO/sea level in the northeastern Atlantic; (b) the same for theNAO/Mediterranean sea level; (c) the same for the SOI/Indian Ocean sea level; (d) the same for the SOI/northeastern Pacific sea level. Contours are wavelet squared coherencies. The vectors indicate the phasedifference (a horizontal arrow pointing from left to right signifies in-phase, and an arrow pointingvertically upward means the second series lags the first by 90 degrees (i.e., the phase angle is 270�)). Inall panels the black thick line is the 5% significance level using the red noise model. Data analyzed areannual sea level.

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possibly larger error source for sea level estimates [Cazenaveand Nerem, 2004]. Our results show that error can bereduced with substantial increase of number of tide gaugerecords; for example, for the period 1800–1900 the errorsare 2–3 mm/yr, compare to 1 mm/yr during 20th century.

5. Conclusion

[33] We developed a new ‘‘virtual station’’ method ofanalyzing sea level records from across different oceanicregions. This is statistically more appropriate than assumingindependent errors from all stations in a region, as there areimportant multiyear cycles in sea level that have differentphase lags at stations across an ocean region. We can thenget an appropriate sea level curve for each region, andimportantly, an error estimate is also produced that can thenbe used in analysis of the nonlinear trend from SSA withconfidence intervals.[34] We have shown that the development of global sea

level rise is highly dependent on the time period chosen andthe global sea level rise occurred during the period from1920 to 1945 is comparable with present-day rate of sealevel rise. Nonlinear sea level trends for the ocean regionsare nonuniform even after 30 year smoothing (trends in highlatitude ocean regions have huge uncertainties, which is achallenge for future study). We provide evidence that 2–13.9 year variability in sea level records is increasing duringthe past 50 years for most of the ocean basins.[35] Our study demonstrates the importance of identify-

ing the effect of multiyear to decadal scale anomalies on sealevel trends in very short time series. We also show thatlow-frequency, multidecadal variability is significant andthat linear trends do not properly represent this. By remov-ing the high frequency variability we reduce the uncertain-ties associated with redistribution of sea level due to varyinggeographical locations of tide gauges along the coastline aswell as noise associated with coastal processes and deter-mine the sea level rise driven by the climatic forcing.[36] We demonstrate that advanced statistical methods

improve error estimations and reduce uncertainties forcalculation of regional and global sea level rise. However,the main source for the uncertainties in sea level studiesusing tide gauge records still remain: poor historical distri-bution of tide gauges, lack of the data from SouthernHemisphere (Africa and Antarctica), the GIA correctionsused, and localized tectonic activity.

[37] Acknowledgments. Some of our software includes code origi-nally written by E. Breitenberger of the University of Alaska adapted fromthe freeware SSA-MTM Toolkit: http://www.atmos.ucla.edu/tcd/ssa. Wethank the Thule Institute and Finnish Academy for financial support; twoanonymous referees provided useful comments.

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�����������������������A. Grinsted and J. C. Moore, Arctic Centre, University of Lapland, 96101

Rovaniemi, Finland.S. Holgate and S. Jevrejeva, Proudman Oceanographic Laboratory,

Liverpool L3 5DA, UK. (sveta@pol.ac.uk)

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